Limiting Eccentricity in Restricted Hierarchical Three-Body Systems with Short-Range Forces

TL;DR

The paper analyzes the maximum inner-binary eccentricity in restricted hierarchical three-body systems under tertiary perturbations and short-range forces, focusing on how quadrupole and octupole terms, GR, and different averaging schemes constrain eccentricity and enable orbital flips. Using the DA framework, it derives analytic expressions for $e_{\max}$ and $e_{\rm lim}$, showing that SRFs cap eccentricity and that octupole terms can drive flips with $e_{\rm flip}$ closely matching $e_{\rm lim}$. It further investigates moderate hierarchies with the Brown Hamiltonian and finds BH shifts in $e_{\max}$ but no change to $e_{\rm lim}$ at $i_0=90^{\circ}$; the SA model yields higher $e_{\max}$, with a transferable analytic estimate for $e_{\rm lim,SA}$. Overall, the results demonstrate that short-range forces impose a universal ceiling on eccentricity even in the presence of strong octupole excitations, while the choice of averaging method governs the precise limiting value and the occurrence of flips. The work provides practical formulas for predicting limiting eccentricities across DA and SA regimes and clarifies the role of BH corrections in moderately hierarchical systems.

Abstract

A hierarchical three-body model can be widely applied to diverse astrophysical settings, from satellite-planet-star systems to binaries around supermassive black holes. The octupole-order perturbation on the inner binary from the tertiary can induce extreme eccentricities and cause orbital flips of the binary, but short-range forces such as those due to General Relativity (GR) may suppress extreme eccentricity excitations. In this paper, we consider restricted hierarchical three-body systems, where the inner binary has a test-mass component. We investigate the maximum possible eccentricity (called "limiting eccentricity") attainable by the inner binary under the influence of the tertiary perturbations and GR effect. In systems with sufficiently high hierarchy, the double averaging (DA) model is a good approximation; we show that the orbits which can flip under the octupole-order perturbation reach the same limiting eccentricity, which can be calculated analytically using the quadrupole-order Hamiltonian. In systems with moderate hierarchy, DA breaks down and the so-called Brown Hamiltonian is often introduced as a correction term; we show that this does not change the limiting eccentricity. Finally, we employ the single averaging (SA) model and find that the limiting eccentricity in the SA model is higher than the one in the DA model. We derive an analytical scaling for the modified limiting eccentricity in the SA model.

Figures (12)

• Figure 1: Level curves of (normalized) Hamiltonian for the "quadrupole + GR" model, with $\epsilon_{\rm GR}=0.02$ and the (normalized) $z$-component of the inner orbital angular momentum $H=0.6$ (see Eq. \ref{['eq_H']}). The color bars represent different values of Hamiltonian $\hat{\cal{H}}={\cal H}/\Phi_0$.
• Figure 2: Top panel: The maximum eccentricity that the inner orbit can achieve as a function of the initial inclination $i_{0}$, with the initial eccentricity set to $e_{0}=0.2$. The black line is for the pure quadrupole model without GR. The red line shows the analytical result given by Eq. (\ref{['equa1']}). The blue line shows the maximum eccentricity with the initial $\omega_0=90^{\circ}$. Both the red and blue lines correspond to the "quadrupole + GR" model with $\epsilon_{\rm {GR}}=0.02$. Bottom panel: The corresponding inclination when the inner orbit reaches $e_{\max}$.
• Figure 3: The maximum eccentricity $e_{\max}$ as a function of initial inclination $i_0$ for different values of $\epsilon_{\rm GR}$, with the initial eccentricity set to $e_0=0.2$ and $\epsilon_{\rm Oct}$ set to 0.02. The red dots are the results with the initial $\omega_0=0$ and $\Omega_0=0$. The black asterisks are the results obtained by scanning $\omega_0$ and $\Omega_0$ in the range $[0,2\pi]$. These results are obtained by integrating Eq. (\ref{['equa_motion']}) for 1500 ZLK cycles for each parameter set.
• Figure 4: The maximum eccentricity as a function of the initial inclination $i_{0}$. The parameters are the same as in Fig. \ref{['figOCT1']}. Each blue dot denotes the maximum eccentricity for a set of initial $\omega_{0}$ and $\Omega_{0}$ in the range $[0,2\pi]$. For a given $i_0$, the highest value of the maximum eccentricities obtained from different initial $\omega_0$ and $\Omega_0$ is marked by a black asterisk. The black asterisks are the same as in Fig. \ref{['figOCT1']}. If the orbit represented by the black asterisk can realize flip, it is additionally marked by a yellow dot. The magenta asterisks are obtained by Eq. (\ref{['equa1']}).
• Figure 5: Top: The initial angles corresponding to the black asterisks in the middle panel in Fig. \ref{['figOCT2']}. Bottom: The angles when the maximum eccentricity is reached. The red dots represent the orbits that can realize flips, and the black dots are orbits that cannot flip.