A Kustaanheimo-Stiefel regularization of the elliptic restricted three-body problem and the detection of close encounters with fast Lyapunov indicators (Q6554941)

In this paper, the Kustaanheimo-Stiefel (KS) regularization is presented, of the elliptic restricted three-body problem (ER3BP), at the secondary body \(P_2\), and the authors discuss its use for the study of a category of transits through its Hill's sphere (fast close encounters). The authors start from the Hamiltonian representation of the problem using the synodic rotating-pulsating reference system and the true anomaly of \(P_2\) as independent variable, and they 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; this translates into an efficient algorithm that can be implemented to numerically integrate the equations of motion. Thus, they constructed a local geometric Kustaanheimo-Stiefel regularization of the spacial ER3BP using a symlectic iso-energetic reduction of the phase-space. They also obtain an algorithm to apply the regularization, efficiently to close encounter solutions. So, using the developed KS formalism, the authors extend the characterization of fast close encounters with the secondary body \(P_2\), from the CR3BP to the ER3BP, by proving their hyperbolic type during the transit in the Hill's sphere for values of μ below a given small limit (here \(m_1=1-\mu\), \(m_2=\mu\) for \(\mu \in (0,1/2)\)), while they do not require a smallness condition on the eccentricity of the primaries. In the numerical part, they test the KS algorithm against a standard Cartesian integration and report the significant advantages in exploiting the regularization, especially in terms of the Hamiltonian conservation in the extended phase-space. For small values of \(\mu\), the authors justify the effectiveness of the regularized fast Lyapunov indicators to detect orbits with multiple fast close encounters. Finally, they provide numerical demonstrations and show the benefits of the regularization in terms of the computational cost.