Quasi-conservative Integration Method for Restricted Three-body Problem

Abstract

The simplest restricted three-body problem, in which two massive points and a massless point particle attract one another according to Newton’s law of inverse squares, has pulsating Hill’s regions where the massless particle moves inside the closed regions surrounding only one of the massive points. Until now, no numerical integrator is known to maintain these regions, making it challenging to reproduce the phenomenon of gravitational capture of massless particles. In this article, we propose a second-order integrator that preserves Hill’s regions to accurately simulate this phenomenon. Our integrator is based on a logarithmic Hamiltonian leapfrog method developed by Mikkola and Tanikawa and features a parameter that is adjusted to preserve a second-order approximation of an invariant integration relation of this restricted three-body problem. We analytically and numerically clarify that this integrator has the following properties: (i) it retains the collinear and triangular Lagrangian solutions regardless of the eccentricity of the relative orbit of the two massive points, (ii) it has the same Hill stability criterion for satellite-type motion of the massless point particle as the original problem, and (iii) it conserves the Jacobi integral for zero eccentricity.