A Dynamical Systems Approach to the Theory of Circumbinary Orbits in the Circular Restricted Problem

TL;DR

This paper uses dynamical systems theory and Floquet analysis to characterize the stability and geometry of circumbinary orbits in the Circular Restricted Three-Body Problem, providing insights into planet formation zones around binary stars.

Contribution

It introduces a Floquet-based framework to analyze stability bifurcations of circumbinary limit cycles and applies this to real systems like Pluto-Charon.

Findings

Identification of an innermost near-circular stable orbit.

Discovery of an exclusion zone inhibiting inward migration of planets.

Validation of analytical results with N-body simulations.

Abstract

To better understand the orbital dynamics of exoplanets around close binary stars, i.e., circumbinary planets (CBPs), we applied techniques from dynamical systems theory to a physically motivated set of solutions in the Circular Restricted Three-Body Problem (CR3BP). We applied Floquet theory to characterize the linear dynamical behavior -- static, oscillatory, or exponential -- surrounding planar circumbinary periodic trajectories (limit cycles). We computed prograde and retrograde limit cycles and analyzed their geometries, stability bifurcations, and dynamical structures. Orbit and stability calculations are exact computations in the CR3BP and reproducible through the open-source Python package pyraa. The periodic trajectories produce a set of non-crossing, dynamically cool circumbinary orbits conducive to planetesimal growth. For mass ratios $\mu \in [0.01, 0.50]$ we found recurring features in the prograde families. These features include: (1) an innermost near-circular trajectory, inside which solutions have resonant geometries, (2) an innermost stable trajectory ($a_{cr} \approx 1.61 - 1.85 \, a_\textrm{bin}$) characterized by a tangent bifurcating limit cycle, and (3) a region of dynamical instability ($a \approx 2.1 \ a_\textrm{bin}; \Delta a \approx 0.1 \ a_\textrm{bin}$), the exclusion zone, bounded by a pair of critically stable trajectories -- bifurcating limit cycles. The exterior boundary of the exclusion zone is consistent with prior determinations of $a_{cr}$ around a circular binary. We validate our analytic results with N-body simulations and apply them to the Pluto-Charon system. The absence of detected CBPs in the inner stable region, between the prograde exclusion zone and $a_{cr}$, suggests that the exclusion zone may inhibit the inward migration of CBPs.