Escape and collision dynamics in the planar equilateral restricted four-body problem

Abstract

We consider the planar circular equilateral restricted four body-problem where a test particle of infinitesimal mass is moving under the gravitational attraction of three primary bodies which move on circular orbits around their common center of gravity, such that their configuration is always an equilateral triangle. The case where all three primaries have equal masses is numerically investigated. A thorough numerical analysis takes place in the configuration (x, y) as well as in the (x, C) space in which we classify initial conditions of orbits into four main categories: (i) bounded regular orbits, (ii) trapped chaotic orbits, (iii) escaping orbits and (iv) collision orbits. Interpreting the collision motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape and the collision basins and we managed to correlate them with the corresponding escape and collision times of orbits. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting dynamical system.

Orbital dynamics in the photogravitational restricted four-body problem: Lagrange configuration

“…In the present paper, we shall study the four-body problem in the Lagrangian configuration by considering the primary body m 1 as the radiation source. Here, we extend the works by Baltagiannis & Papadakis [11], Zotos [18], and Papadouris & Papadakis [20], by considering not only three different combinations of mass for the primary bodies (three equal masses, two equal masses, and three different masses), but also the radiation pressure. Moreover, taking into account that in our Solar system the Sun, Jupiter and the Trojan asteroids form an equilateral triangle configuration, we will consider the characteristic values of this system for the case of three different masses.…”

“…The Euler and Lagrange configurations of the PR4BP have been widely studied in the literature, ranging from the calculation of the equilibrium points and their stability [10][11][12] to the computation of families of periodic orbits [13][14][15][16][17], or the study of the orbital dynamics of escape and collision in these systems [18,19]. Over the years, several modifications to the basic Euler and Lagrange configurations have been investigated to understand the influence and effects of different parameters in realistic celestial systems.…”

On the use of surfaces of section in the N‐body ring problem

“…In 1969, Hénon introduced the section , to study the restricted three‐body problem. This plane has been also used by authors like Zotos, in his numerical exploration of the four‐body problem, and Barrio et al, in the analysis of the fractal structures in the Hénon‐Heiles potential . In this new plane, the initial conditions of the orbits are taken in the x axis, with x = x 0 , y =0, and with the initial velocity parallel to the y axis ( , ).…”

Divulging the effect of small perturbations in the Coriolis and centrifugal forces in the photogravitational version of autonomous restricted four‐body problem with oblate primary

“…They have shown that the perturbations in these forces have a substantial impact on the regions of possible motion of the fourth body. Zotos () has discussed the escape and collision dynamics and the Newton–Raphson basins of convergence in the planar equilateral restricted four‐body problem, respectively.…”