Secular dynamics of binaries in stellar clusters – I. General formulation and dependence on cluster potential

Abstract

Orbital evolution of binary systems in dense stellar clusters is important in a variety of contexts: origin of blue stragglers, progenitors of compact object mergers, millisecond pulsars, and so on. Here we consider the general problem of secular evolution of the orbital elements of a binary system driven by the smooth tidal field of an axisymmetric stellar cluster (globular, nuclear, etc.) in which the binary orbits. We derive a secular Hamiltonian (averaged over both the inner Keplerian orbit of the binary and its outer orbit within the cluster) valid to quadrupole order for an arbitrary cluster potential and explore its characteristics. This doubly-averaged 'tidal' Hamiltonian depends on just two parameters, which fully absorb the information about the background cluster potential and the binary's orbit within it: a dimensional parameter A setting the secular timescale, and a dimensionless parameter Γ which determines the phase portrait of the binary's inner orbital evolution. We examine the dependence of A and Γ on cluster potential (both spherical and axisymmetric) and on the binary orbit within the cluster. Our theory reproduces known secular results -such as Lidov-Kozai evolution and the effect of the Galactic tide on Oort Cloud comets -in appropriate limits, but is more general. It provides a universal framework for understanding dynamical evolution of various types of binaries driven by the smooth tidal field of any axisymmetric potential. In a companion paper (Hamilton & Rafikov 2019b) we provide a detailed exploration of the resulting orbital dynamics.

Compact-object Mergers in the Galactic Center: Evolution in Triaxial Clusters

“…In this section, we examine the dynamics of binaries whose outer orbits densely fill an axisymmetric torus on timescales shorter than those at which the inner binary is torqued by the tidal field. This problem was studied in detail by Hamilton & Rafikov (2019b, 2019c, who develop a general formalism to describe the dynamics of torus-filling binaries in axisymmetric potentials, generalizing previous results regarding the effect of galactic tides on stellar binaries (e.g., Heisler & Tremaine 1986). Here, we demonstrate that our simulations are consistent with the results of Hamilton & Rafikov (2019b, 2019c and examine the additional effect of an SMBH on these considerations.…”

“…At larger a out , the influence of the cluster is more apparent, causing a dip in the value of Γ. However, the presence of the SMBH keeps Γ greater than the critical value of 1/5 at all distances, in contrast to the behavior seen in Hamilton & Rafikov (2019b). At large values of a out , Γ begins to converge again toward unity, as expected.…”

“…In this section, we discuss the dynamics of binaries in axisymmetric potentials whose outer orbits fail to densely fill a torus within the secular timescale. In this regime, the analytic formalism of Hamilton & Rafikov (2019b, 2019c breaks down, and we observe substantially different behavior. Most interestingly, this regime can give rise to secular chaos in the evolution of the inner orbit, as previously studied by Petrovich & Antonini (2017).…”

“…As shown by Hamilton & Rafikov (2019b), in this torusaveraged limit, the tidal tensor has only diagonal terms and áF ñ = áF ñ…”

“…However, the authors treated the cluster potential as a small perturbation to a three-body system and did not explore the full extent of the cluster, but rather limited themselves to the central ∼0.5 pc of a single ad hoc potential. Second, in a series of papers, Hamilton & Rafikov (2019a, 2019b, 2019c explored the secular dynamics for the full extent of the cluster and overcame the technical limitations of Petrovich & Antonini (2017) by fully accounting for the cluster tidal field in the binary's evolution. However, the authors restricted themselves to non-triaxial clusters without an SMBH.…”