Capture in the Circular and Elliptic Restricted Three-Body Problem

Abstract

In this paper the authors provide a study of the phenomenon of the gravitational capture by using the models of the circular and elliptic restricted three-body problem. In the first part the inadequacy of the circular restricted three-body problem in the study of the phenomenon of the capture in the case of NEAs is shown. In the model of the spatial elliptic restricted three-body problem criteria of the capture are deduced by using the pulsating Hill-regions.

About the Hill stability of extrasolar planets in stellar binary systems

“…The variation of the zero velocity surfaces is proved by using a three-dimensional invariant relation deduced earlier (Makó and Szenkovits 2004). In this model the zero velocity surfaces are changing their topological type, and even closed Hillspheres can open.…”

“…where v is the velocity of the third massless particle (Makó and Szenkovits 2004). For a given set of initial conditions…”

The three-body problem

“…The fact that there are ejection-collision orbits in the 3D ER3BP was demonstrated by Llibre and Pinol [1990]. Moreover, the effectiveness of the phenomenon of the gravitational capture of the third body by the primaries in the 3D ER3BP with small µ was discussed by Makó and Szenkovits [2004]. Sinclair [1970] identified and classified various families of periodic orbits in the commensurable planar ER3BP, where two bodies are close to commensurability in their mean motions taking place in the same plane around the central body.…”

Analytical criteria of Hill stability in the elliptic restricted three body problem

“…At present, the Hill stability criteria in the elliptic restricted three body problem is not well developed due to the timedependent Jacobi integral. Mako (2004Mako ( , 2005Mako ( , 2014, Mako and Szenkovits (2008), Mako et al (2010) have done a lot of work on this topic. A similar criterion to the circular restricted three body problem is given to measure the Hill stability .…”