Analytic computation of the secular effects of encounters on a binary: features arising from second-order perturbation theory

Abstract

Binary-single interactions play a crucial role in the evolution of dense stellar systems such as globular clusters. In addition, they are believed to drive black hole (BH) binary mergers in these systems. A subset of binary-single interactions are secular encounters, for which the third body approaches the binary on a relatively wide orbit, and such that it is justified to average the equations of motion over the binary's orbital phase. Previous works used firstorder perturbation theory to compute the effects of such secular encounters on the binary. However, this approach can break down for highly eccentric binaries, which are important for BH binary mergers and gravitational wave sources. Here, we present an analytic computation using second-order perturbation techniques, valid to the quadrupole-order approximation. In our calculation, we take into account the instantaneous back-reaction of the binary to the third body, and compute corrections to previous first-order results. Using singly-averaged and direct 3-body integrations, we demonstrate the validity of our expressions. In particular, we show that the eccentricity change for highly eccentric binaries can reach a plateau, associated with a large inclination change, and can even reverse sign. These effects are not captured by previous first-order results. We provide a simple script to conveniently evaluate our analytic expressions, including routines for numerical integration and verification. M R Binary j e Perturber Eccentricity E Periapsis distance Q CM θ x z Figure 1. Sketch of the configuration. A binary (center of mass 'CM') is perturbed by a passing object (mass M) on a parabolic or hyperbolic orbit (eccentricity E) with periapsis distance Q to the binary center of mass (Q > a, where a is the binary's semimajor axis).by Heggie & Rasio (1996) are commonly used to take into account the effects of (distant) encounters on a binary in Monte Carlo-style computations (e.g., Rasio & Heggie 1995; Spurzem et al. 2009;Geller et al. 2019).However, there are situations in which the first-order perturbation approach can break down. In Heggie & Rasio (1996), it was discussed that their method no longer applies if the eccentricity change is of the order of the initial eccentricity, i.e., ∆e ∼ e. However, as we will show here, when the binary is already highly eccentric, then the first-order approach can break down even if ∆e is small, in particular, if ∆e ∼ 1 − e. The latter can occur in highly eccentricity binaries perturbed by distant (i.e., weak) encounters, which can drive relatively large changes in the binary's orbital angular momentum. A binary can be driven to high eccentricities as a result of strong (non-secular) encounters in dense stellar systems. The breakdown of the first-order method originates from the fact that the binary's elements can change significantly during the passage of the perturber. Therefore, it is not justified to assume that these elements are constant when integrating the equations of motion.In this paper, we present an analytic compu...

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Structures of secular resonances for inner test particles in hierarchical planetary systems

“…To make sure that the double-averaged Hamiltonian could effectively predict the long-term evolution of the test particle, it is required that the mass of the perturber should be much smaller than that of the central body. Otherwise, the effects filtered by double averaging need to be taken into account in formulating the dynamical model of long-term evolution ( Ćuk and Burns 2004;Luo et al 2016;Lei et al 2018;Lei 2019;Hamers and Samsing 2019).…”

Structures of secular resonances for inner test particles in hierarchical planetary systems

“…To make sure that the double-averaged Hamiltonian could effectively predict the long-term evolution of the test particle, it is required that the mass of the perturber should be much smaller than that of the central body. Otherwise, the effects filtered by double averaging need to be taken into account in formulating the dynamical model of long-term evolution ( Ćuk and Burns 2004;Hamers and Samsing 2019;Lei 2019;Lei et al 2018;Luo et al 2016).…”