The unreasonable effectiveness of the $n \Sigma v$ approximation

TL;DR

This paper investigates the accuracy of the $n \\Sigma v$ approximation in dense stellar systems, revealing it often fails in idealized potentials but generally works due to orbital precession in realistic environments.

Contribution

The study demonstrates that the $n \\Sigma v$ approximation's validity depends on orbital dynamics, showing it can significantly deviate in idealized potentials but is usually recovered in real systems.

Findings

Interaction rates in ideal potentials depend on orbital intersections.

Orbital precession restores the $n \\Sigma v$ approximation in most cases.

Failures occur mainly near intermediate-mass black holes.

Abstract

In kinetic theory, the classic $n \Sigma v$ approach calculates the rate of particle interactions from local quantities: the number density of particles $n$, the cross-section $\Sigma$, and the average relative speed $v$. In stellar dynamics, this formula is often applied to problems in collisional (i.e. dense) environments such as globular and nuclear star clusters, where blue stragglers, tidal capture binaries, binary ionizations, and micro-tidal disruptions arise from rare close encounters. The local $n \Sigma v$ approach implicitly assumes the ergodic hypothesis, which is not well motivated for the densest star systems in the Universe. In the centers of globular and nuclear star clusters, orbits close into 1D ellipses because of the degeneracy of the potential (either Keplerian or harmonic). We find that the interaction rate in perfectly Keplerian or harmonic potentials is determined by a global quantity -- the number of orbital intersections -- and that this rate can be far lower or higher than the ergodic $n \Sigma v$ estimate. However, we find that in most astrophysical systems, deviations from a perfectly Keplerian or harmonic potential (due to e.g. granularity or extended mass) trigger sufficient orbital precession to recover the $n \Sigma v$ interaction rate. Astrophysically relevant failures of the $n \Sigma v$ approach only seem to occur for tightly bound stars orbiting intermediate-mass black holes, or for the high-mass end of collisional cascades in certain debris disks.