# Analytic computation of the secular effects of encounters on a binary: third-order perturbation, octupole, and post-Newtonian terms; steady-state distribution

## TL;DR

This paper extends analytical models of secular encounters affecting binary black holes in globular clusters by including third-order perturbations, octupole terms, and post-Newtonian effects, providing a comprehensive framework for cluster evolution simulations.

## Contribution

It introduces third-order perturbation theory and octupole terms into the analysis of secular encounters, enhancing the accuracy of binary evolution models in dense stellar systems.

## Key findings

- Derived third-order perturbation expressions for eccentricity and angular momentum.
- Included octupole and post-Newtonian terms in the analytical framework.
- Calculated steady-state distribution using angular-momentum diffusion coefficients.

## Abstract

Dense stellar systems such as globular clusters are believed to harbor merging binary black holes (BHs). The evolution of such binaries is driven by interactions with other stars, most notably, binary-single interactions. Traditionally, so-called `strong' interactions are believed to be the driving force in this evolution. However, we recently showed that more distant, i.e., "weak" or "secular" encounters, can have important implications for the properties of merging BH binaries in globular clusters. This motivates more detailed understanding of the effects of secular encounters on a binary. In another previous paper, we analytically calculated expressions for the changes of the eccentricity and angular-momentum vectors taking into account second-order (SO) perturbation theory, and showed that, for highly eccentric binaries, the new expressions give rise to behavior that is not captured by first-order (FO) theory. Here, we extend our previous work to third order (TO) perturbation theory. We also include terms up to and including octupole order. The latter are nonzero for binaries with unequal component masses. In addition, we consider the effects of post-Newtonian terms, and we determine the steady-state distribution due to the cumulative effect of secular encounters by computing the associated angular-momentum diffusion coefficients, and applying the Fokker-Planck equation. Together with our previous work, the results in this paper provide a framework for incorporating the effects of distant encounters on binaries in models of cluster evolution, such as Monte Carlo codes.