[Paper Review] Small-N Collisional Dynamics V: Beyond the Realm of Not-So-Small-N

Carlos Barrera, Nathan W. C. Leigh+3 more|arXiv (Cornell University)|Oct 30, 2020

Stellar, planetary, and galactic studies2 references4 citations

TL;DR

This study tests the validity of the mean free path approximation for predicting stellar collision rates in dense star clusters by comparing analytic models to direct N-body simulations across particle counts from N ≈ 10 to N ≈ 1000. The model, based on Leigh et al. (2017), shows excellent agreement with simulations—typically within one standard deviation—demonstrating its robustness for similar particle types and moderate N, but with diminishing accuracy at higher N due to relaxation time effects and unaccounted cluster evolution processes.

ABSTRACT

Direct collisions between finite-sized particles occur commonly in many areas of astrophysics. Such collisions are typically mediated by chaotic, bound gravitational interactions involving small numbers of particles. An important application is stellar collisions, which occur commonly in dense star clusters, and their relevance for the formation of various types of stellar exotica. In this paper, we return to our study of the collision rates and probabilities during small-number chaotic gravitational interactions ($N$ $\lesssim$ 10), moving beyond the small-number particle limit and into the realm of larger particle numbers ($N$ $\gtrsim$ 10$^3$) to test the extent of validity of our analytic model as a function of the particle properties and the number of interacting particles. This is done using direct $N$-body simulations of stellar collisions in dense star clusters, by varying the relative numbers of particles with different particle masses and radii. We compute the predicted rate of collisions using the mean free path approximation, adopting the point-particle limit and using the sticky-star approximation as our collision criterion. We evaluate its efficacy in the regime where gravitational-focusing is important by comparing the theoretical rates to numerical simulations. Using the tools developed in previous papers in this series, in particular Collision Rate Diagrams, we illustrate that our predicted and simulated rates are in excellent agreement, typically consistent with each other to within one standard deviation.

Motivation & Objective

• To test the validity of the mean free path approximation for stellar collision rates in dense star clusters beyond the small-N regime (N ≤ 10), extending into the large-N regime (N ≳ 10^3).
• To evaluate the performance of the analytic model from Leigh et al. (2017) in predicting collision rates across varying particle mass and radius distributions.
• To identify the critical particle number at which the assumptions of the small-N analytic model begin to break down due to relaxation time effects and cluster evolution.
• To assess the efficacy of the sticky-sphere approximation and mean free path formalism in regimes where gravitational focusing is significant.
• To provide a foundation for replacing computationally expensive N-body integrators with analytic collision probability models in Monte Carlo cluster evolution simulations.

Proposed method

• Conduct direct N-body simulations of stellar systems with N ≈ 1000 particles, using Plummer distributions for initial conditions and varying particle masses and radii.
• Apply the sticky-sphere approximation as the collision criterion, where colliding particles are treated as fully merging upon contact.
• Compute theoretical collision rates using the mean free path approximation: R_coll ≈ nσ_collv, with σ_coll derived from gravitational focusing.
• Use Collision Rate Diagrams to visualize and compare predicted vs. simulated collision rates across different particle types and spatial regions.
• Compare analytic predictions from Leigh et al. (2017) to simulated collision counts, evaluating agreement within one standard deviation.
• Vary system parameters including mass ratios, size ratios, and total particle counts to probe model robustness across diverse astrophysical conditions.

Experimental results

Research questions

• RQ1 How well does the mean free path approximation predict collision rates in large-N stellar systems (N ≳ 1000) compared to direct N-body simulations?
• RQ2 At what particle number does the small-N analytic model from Leigh et al. (2017) begin to break down due to relaxation time effects and cluster evolution?
• RQ3 How do mass and radius ratios between particle types affect the accuracy of the analytic collision rate model?
• RQ4 To what extent does gravitational focusing influence the agreement between analytic predictions and simulated collision rates in dense clusters?
• RQ5 Can the analytic model be reliably used to replace small-number gravity integrators in Monte Carlo cluster evolution codes?

Key findings

• The analytic model from Leigh et al. (2017) shows excellent agreement with N-body simulations, with predicted and simulated collision rates typically consistent within one standard deviation.
• For equal-mass particles, the model predicts collision rates within a factor of ~5 of simulated values when considering only core collisions, but this offset is corrected when integrating over the entire cluster volume.
• The model performs best for similar particle types (small mass and size ratios), as it does not account for mass segregation or cluster evolution effects observed in simulations.
• Performance degrades at higher N (≳1000), likely due to longer relaxation times compared to collision times, which invalidates time-averaged assumptions in the small-N model.
• The model remains valid for predicting collision probabilities during individual few-body interactions (N = 3–6), suggesting potential for replacing N-body integrators in Monte Carlo cluster evolution codes.
• Future work will identify the critical N at which the transition from small-N to shell-based Monte Carlo modeling becomes necessary for accurate cluster evolution simulations.