On the exploration of the (N + 2)‐body ring problema
In our current research, we have conducted a comprehensive analysis of the planar and three-dimensional (N + 2)-body ring problem, exploring various aspects such as the evolution of zero-velocity curves depending the values of the mass ratio, critical values of the Jacobi constant, and their impact in the dynamics of escape orbits in three dimensions. Our investigation into the planar (N + 2)-body ring problem with N = 3 to 100 peripheral bodies, and the presence of a central mass (β > 0), reveals key insights into the behavior of zero-velocity curves. For this purpose, we have accomplished a very accurate analysis of the different intersections of the so-called critical values of the Jacobi constant, which characterize the sets of stationary points of the system, when β is lower than the bifurcation value of the system, β∗N . Through detailed analysis, we have determined that up to seven distinct intersections of these critical values can occur, each corresponding to a specific β value, here denoted as βEi , i = 1, 2, . . . , 7. The arrangement and evolution of these intersections are influenced by the number of peripheralbodies, with significant implications for the system’s stationary points. Notably,the absence of intersections between zones A2 and D1 when N > 16 indicates that isolated islands around the central mass persist even as the zero-velocity curves open. Due to the fact that the set {z = 0, z˙ = 0} is invariant in this system, the results of this investigation can be applied to explain the motion of an infinitesimal particle that lies on the XY plane in the 3D (N +2)−body ring problem, which we can then conclude that will be subject to the same structure of the zero-velocity curves in the plane. Additionally, we have presented a numerical integration by means of recurrent power series method, adapting it to address the three-dimensional (N +2)-body ring problem. This method has proven both accurate and efficient, enabling the computation of the intricate motion of the test particle in the threedimensional space, while preserving the system’s inherent complexity. In the context of the three-dimensional (N +2)-body ring problem, we have examined the influence of the mass ratio, β, and the Jacobi constant, C, on the escape dynamics with N = 4 peripheral bodies, considering a new surface of section in accordance to the symmetries of the problem. The integration has been performed on a set of initial conditions selected on the boundary of a sphere of radius ρ = 0.3 centered at the origin of the system, with a grid of 2048 × 2048 points uniformly distributed in the hyperplane defined by (ϕ, θ) ∈ [0, π] × θ ∈ [0, 2π), and such that the initial velocity at each initial position is perpendicular to the tangent plane of the sphere at that point. This way of selecting the initial conditions allows to explore in a two-dimensional surface the basins of escape of a six-dimensional system. In order to explore the intricate influence of the mass ratio, β, and the Jacobi constant, C, on the system’s overall dynamics and the geometry of escape, we have made an analysis for the values β = 2, 10, 20. For each of these values, we have examined four corresponding values of C such that the zero-velocity surfaces are open, with four exit windows of approximately equal size. The main conclusions that can be arisen from this research are: • The initial conditions that lead to escaping orbits are the ones that lie on the vicinity of the equator, regardless of the value of β. • When β increases, the subset of initial conditions of escaping orbits is reduced in a particular pattern: this domain starts to be split up into several disconnected regions of escape, in particular, there are four regions for β = 10 and eight regions for β = 20. • The structure of the basins of escape along the sphere is symmetrical with respect to the horizontal plane and to four planes that are perpendicular to Oxy. • The distribution of the escaping orbits along the equator is interrelated and it can be arranged based on a counterclockwise tour along the exit windows in the (x, y, z) configuration space, beginning from the first exit channel. • There is a distinct morphological transformation in the behavior of the basins of escape that depends on the parameters β and C. • As the value of β increases from 2 to 10, there is a change in the shapes of the basins of escape, transitioning from full-shaped semi-ellipses to rings that have the same color as the facing window. In addition, when β = 20, a significant alteration in both the form and the size of the basins of escape is observed, consisting of nested rings that gradually shrink from a semi-ellipse shape to a crescent shape. • The effect of the parameter C is notably significant, either leading to the emergence of new shapes or to the shrinking and expansion of existing ones due to the increase in the window’s width. The results found on the behaviour of the geometry of escape depending on the values of β analyzed here, seem to evidence the existence of critical values of this parameter such that the domains of escape reduce and split up into pairs, quartets, sextets, etc., which indeed may be related to the symmetries of the (N + 2)-body ring problem. As we conclude, both the three-dimensional and planar analysis presented in this manuscript demonstrate that the fundamental influence of the mass ratio, β and the Jacobi constant, C, remains consistent across dimensions. Specifically, these parameters govern the evolution of zero-velocity curves and have a critical role in escape dynamics and the probability of escape. Future research will focus on examining additional β values to determines the critical β values at which the domains of escape begin to split into pairs, quartets, sextet, octets,etc and also for more accuracy, it is possible to conduct an extensive examination that incorporates a larger range of mass ratio values and the Jacobi constant in order to determine the extent to which the behavior of escape probability is dependent on those parameters. Such sorts of endeavors have the potential to advance the field of celestial mechanics and reveal more profound explanations for the dynamic of orbital systems. Furthermore, we have elucidated the impact of the Jacobi constant, C, and the mass ratio,β, on the probability of escape orbits in the (4+2)-body ring problem in three dimensions, using the same methodology and structure utilized in the previous research. The results that we have obtained reveal that when we consider the Jacobi constant values associated with the opening that has the largest diameter, the probability of escape varies monotonically with respect to the mass ratio, conversely, with the opening that has the smallest diameter, the probability does not vary monotonically. Furthermore, significant variations take place in the distribution of the escaping orbits and the initial time of escape, as the mass ratio increases. It is evident, based on our findings, that the four windows can be considered equiprobable. While there are minor errors due to the numerical integration, the probabilities of escape through each window exhibit a substantial level of similarity in the interval time of fast escape, regardless their unequal distribution in the other intervals. As we conclude, both the three-dimensional and planar analysis presented in the manuscript demonstrate that the fundamental influence of the mass ratio, β and the Jacobi constant, C, remains consistent across dimensions. Specifically, these parameters govern the evolution of the zero-velocity curves and surfaces have a critical role in the escape dynamics and the probability of escape. Future research will focus on examining additional β values to determine the critical β values at which the domains of escape begin to split into pairs, quartets, sextet, octets, etc., and also for more accuracy, it is possible to conduct an extensive examination that incorporates a larger range of mass ratio values and the Jacobi constant in order to determine the extent to which the behavior of escape probability is dependent on those parameters. Such sorts of endeavors have the potential to advance the field of celestial mechanics and reveal more profound explanations for the dynamic of orbital systems.
N-body ring problem, Zero-velocity curves, Celestial Mechanics, Basins of escape, Surface of section, Numerical integration, Probability of escape
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