On the analysis of the fractal basins of escape in the N ‐body ring problem

Abstract

This article summarizes the results of a numerical investigation of the phenomenon of escape in the N-body ring configuration, focusing on the scenarios that result for N = 5, 6, 7, 8 peripheral bodies. There is a critical value of the Jacobi constant of the system such that for smaller values, the potential well opens and test particles may leave the potential through any of its N openings.By means of a surface of section, we show the results of the computation of the basins of escape towards the different directions of escape, analyzing the structures that appear in them.

Analysis of the escape in systems with four exit channels

“…The other four bodies, all of them with the same mass m, are situated at the vertexes of a regular polygon that rotates on its own plane around the center of mass with a uniform angular velocity. 14,15,20 The dimensionless equations of motion describing the motion in a plane of a test particle under the gravitational influence of this system are given by…”

“…The analysis of the escape of a test particle from a dynamical system is an active field of research to which many scientists are contributing nowadays. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] The mechanism that explains how the escape from a system takes place is well known: The limiting curve of the basins of escape in a proper surface of section is determined by projecting the stable manifolds to the unstable periodic orbits that are located at the openings of the curves of zero velocity of the system on this surface. However, it is necessary to calculate these limiting curves in each specific system in order to unveil the properties of the escape.…”

Effect of the mass ratio on the escape in the 4-body ring problem

“…The number of escaping orbits increases to 35 orbits at t 9 and, then, it descends until 2 escaping orbits at t 10. In the intervals (10,13] and (15,20], the escape occurs in a sequential manner. In the interval (13,15], the number of escaping orbits in each of the 10 intervals we have considered is bounded between 4 and 6 orbits.…”

“…The analysis of the geometric structures that govern the escape of a particle from an open Hamiltonian system has focused the efforts of many research groups over the last decades [1-8, 10, 11, 13-22, 24-26]. One of the systems that has aroused much of that interest is a particular case of the N-body problem: the N-body ring problem [18][19][20][21][22]. This attention is due to the fact that this system models a wide range of celestial systems, like planetary rings, asteroid belts and some type of formations of stars and planets, to cite some examples.…”

“…Also, Navarro and Martínez-Belda [20] carry out a numerical exploration of the N-body ring problem in order to analyze the distribution of times of escapes, for N 5, 6, 7, 8 primaries, and considering a fixed value of the mass ratio β 2. Moreover, they fulfill a detailed calculation of the basins of escapes for these four values of N. Navarro et al [22] try to find out common properties in the escape from open Hamiltonian systems with the same number of openings in the potential well.…”