A Kustaanheimo-Stiefel regularization of the elliptic restricted three-body problem and the detection of close encounters with fast Lyapunov indicators

TL;DR

This paper introduces a Kustaanheimo-Stiefel regularization method for the elliptic restricted three-body problem, enabling efficient detection of close encounters via fast Lyapunov indicators, especially for small mass ratios.

Contribution

It develops a KS regularization framework for the ER3BP at the secondary body, facilitating the study of fast close encounters without requiring small eccentricity.

Findings

Regularized Hamiltonian effectively detects fast close encounters.

Solutions exhibit exponential growth during transits, validating the use of RFLIs.

Regularization reduces computational cost in numerical simulations.

Abstract

We present the Kustaanheimo-Stiefel (KS) regularization of the elliptic restricted three-body problem (ER3BP) at the secondary body $P_2$, and discuss its use to study a category of transits through its Hill's sphere (fast close encounters). Starting from the Hamiltonian representation of the problem using the synodic rotating-pulsating reference frame and the true anomaly of $P_2$ as independent variable, we perform the regularization at the secondary body analogous to the circular case by applying the classical KS transformation and the iso-energetic reduction in an extended 10-dimensional phase-space. Using such regularized Hamiltonian we recover a definition of fast close encounters in the ER3BP for small values of the mass parameter $\mu$ (while we do not require a smallness condition on the eccentricity of the primaries), and we show that for these encounters the solutions of the variational equations are characterized by an exponential growth during the fast transits through the Hill's sphere. Thus, for small $\mu$, we justify the effectiveness of the regularized fast Lyapunov indicators (RFLIs) to detect orbits with multiple fast close encounters. Finally, we provide numerical demonstrations and show the benefits of the regularization in terms of the computational cost.