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

Abstract

ABSTRACTDense stellar systems such as globular clusters are believed to harbour 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 perturbation theory, and showed that, for highly eccentric binaries, the new expressions give rise to behaviour that is not captured by first-order theory. Here, we extend our previous work to third-order perturbation theory. We also include terms up to and including octupole order. The latter are non-zero 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.