A free-fall-based switching criterion for P^3 T N-body methods in collisional stellar systems

TL;DR

The paper addresses the challenge of robustly switching between fast approximate long-range gravity and precise short-range N-body solvers in P^3T simulations of collisional stellar systems. It introduces a free-fall-based switching criterion, Δt_soft = P_in / n_s with P_in = 2π sqrt(r_in^3/(G(m_1+m_2))) and extends it to multi-mass systems via Δt_soft = (2π/n_s) sqrt( max(m_i,m_j)/⟨m⟩ · r_in^3 /(G (m_i+m_j)) ), comparing it to the traditional σ-based rule Δt_soft = α r_in / σ. Through extensive simulations with the petar code across equal-mass and IMF clusters, primordial binaries, varying virial ratios, and fractal initial conditions, the study shows the free-fall criterion tends to yield superior energy conservation in low-σ, subvirial, or binary-rich systems, while the σ-based criterion is more accurate for high-σ systems; near-virial, both criteria can perform comparably. The findings highlight that the two criteria probe different physics—mutual gravity vs global potential—and suggest a hybrid approach that leverages the strengths of each across dynamical regimes, with practical guidance on optimal Δt_soft and r_in to balance accuracy and performance across diverse stellar systems.

Abstract

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 free-fall-based switching criterion for general stellar systems, alongside the commonly used velocity-dispersion-based ($σ$-based) criterion. Using the \textsc{petar} code with the P$^3$T scheme and slow-down 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 free-fall-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 free-fall-based criterion improves as the tree timestep is reduced, whereas the $σ$-based degrades due to its low-accuracy treatment of two-body encounters.

Figures (22)

• Figure 1: Free-fall-based switching criterion: set $\Delta t_{\mathrm{soft}}$ to $1/n_\mathrm{s}$ of the circular binary period with semi-major axis $r_{\mathrm{in}}$.
• Figure 2: Eccentric binary orbits shown for various $r_{\mathrm{in}}$.
• Figure 3: Relative cumulative energy error $\epsilon$ for the binary with $m_1/m_2=0.1$ and $\Delta t_{\mathrm{soft}}=0.0078125$ Myr at three $r_{\mathrm{in}}$ values: 0.0005 (pure leapfrog), 0.016 (mixture) and 0.256 (pure Hermite). Simulations start at the apocenter and reach pericenter at about 0.446 Myr. The gray band marks the interval in the mixture scheme when the integration traverses the changeover region (binary separation $r<r_{\mathrm{out}}$).
• Figure 4: The maximum error of energy ($\epsilon_{\mathrm{max}}$) is shown as a function of $\Delta t_{\mathrm{soft}}$ (left) and $n_\mathrm{s}$ (right). Colors represent different $r_{\mathrm{in}}$ in units of apo-center separation $r_{\mathrm{apo}}$, while line styles indicate the two binary mass ratios.
• Figure 5: Similar to Figure \ref{['fig:dtrinbinary']}, but replacing $\epsilon_{\mathrm{max}}$ by $\epsilon_{\mathrm{f}}$.