On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem

Dubeibe, F. L.; Lora-Clavijo, F. D.; González, Guillermo

doi:10.1007/s10509-017-3076-1

Cited by 26 publications

(30 citation statements)

References 30 publications

(38 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For simplicity, in all that follows we shall use canonical units, such that the sum of the masses, as well as the distance between the primaries, the angular velocity, and the gravitational constant, are set to 1. Additionally, as we consider the non-relativistic limit of the model, the speed of light will be chosen to the value c = 1 × 10 4 [19,20], unless otherwise is specified. Taking into account the previous definitions, the equations of motion in a synodic frame of reference r = (x, y) read as…”

Section: Equations Of Motionmentioning

confidence: 99%

Equilibrium Points and Basins of Convergence in the Triangular Restricted Four-Body Problem with a Radiating Body

Osorio-Vargas

González

Dubeibe

2020

Int. J. Bifurcation Chaos

Self Cite

View full text Add to dashboard Cite

The dynamics of the four-body problem have attracted increasing attention in recent years. In this paper, we extend the basic equilateral four-body problem by introducing the effect of radiation pressure, Poynting-Robertson drag, and solar wind drag. In our setup, three primaries lay at the vertices of an equilateral triangle and move in circular orbits around their common center of mass. Here, one of the primaries is a radiating body and the fourth body (whose mass is negligible) does not affect the motion of the primaries. We show that the existence and the number of equilibrium points of the problem depend on the mass parameters and radiation factor. Consequently, the allowed regions of motion, the regions of the basins of convergence for the equilibrium points, and the basin entropy will also depend on these parameters. The present dynamical model is analyzed for three combinations of mass for the primaries: equal masses, two equal masses, different masses. As the main results, we find that in all cases the libration points are unstable if the radiation factor is larger than 0.01 and hence able to destroy the stability of the libration points in the restricted four-body problem composed by Sun, Jupiter, Trojan asteroid and a test (dust) particle. Also, we conclude that the number of fixed points decreases with the increase of the radiation factor.

show abstract

Section: Equations Of Motionmentioning

confidence: 99%

Equilibrium Points and Basins of Convergence in the Triangular Restricted Four-Body Problem with a Radiating Body

Osorio-Vargas

González

Dubeibe

2020

Int. J. Bifurcation Chaos

Self Cite

View full text Add to dashboard Cite

show abstract

“…A series of research papers (e.g., [23]; [12]; [14]) are available on the 1 st order post-Newtonian equations of motion for the restricted problem of three bodies which are deduced by using the Einstein-Infeld-Hoffmann theory (e.g., [19], [18]). [17] studied the dynamics of the planar circular restricted problem of three-bodies in the context of a pseudo-Newtonian approximation by using the Fodor-Hoenselaers-Perjé procedure, while [20] examined the influence of the separation between the primaries.…”

Section: Introductionmentioning

confidence: 99%

“…When the value of the transition parameter ∈ (0, 0.67752839], the libration points L 11,12,13,14 , L 15,16,17,18 and L 19,20,21,22 move away from the centers of the primaries P 1 , P 2 , and P 3 , respectively, while on the other hand the libration points L 2,7,8 move towards the centers of the primaries P 1 , P 2 , and P 3 , respectively. In particular, L 1,3,5,6,9 and L 10 , move towards the central libration point L 4 .…”

mentioning

confidence: 99%

Unveiling the basins of convergence in the pseudo-Newtonian planar circular restricted four-body problem

et al. 2019

View full text Add to dashboard Cite

The dynamics of the pseudo-Newtonian restricted four-body problem has been studied in the present paper, where the primaries have equal masses. The parametric variation of the existence as well as the position of the libration points are determined, when the value of the transition parameter ∈ [0, 1]. The stability of these libration points has also been discussed. Our study reveals that the Jacobi constant as well as transition parameter have substantial effect on the regions of possible motion, where the fourth body is free to move. The multivariate version of Newton-Raphson iterative scheme is introduced for determining the basins of attraction in the configuration (x, y) plane. A systematic numerical investigation is executed to reveal the influence of the transition parameter on the topology of the basins of convergence. In parallel, the required number of iterations is also noted to show its correlations to the corresponding basins of convergence. It is unveiled that the evolution of the attracting regions in the pseudo-Newtonian restricted four-body problem is a highly complicated yet worth studying problem.

show abstract

“…In [5], Huang and Wu studied the relativistic effect of the separation between two primaries on the dynamics by applying scaling transformations to distance and time; these authors used the equations derived by Maindl and Dvorak [1]. To numerically preserve the generalized Jacobian constant, the authors of [6] presented an alternative system for the PN circular restricted three-body problem using the Einstein-Infeld-Hoffmann (EIH) formalism up to the 1PN order. Furthermore, the authors in [6] claimed that c = 10 4 is suitable for the case of a = 1 in the solar system.…”

Section: Introductionmentioning

confidence: 99%

“…To numerically preserve the generalized Jacobian constant, the authors of [6] presented an alternative system for the PN circular restricted three-body problem using the Einstein-Infeld-Hoffmann (EIH) formalism up to the 1PN order. Furthermore, the authors in [6] claimed that c = 10 4 is suitable for the case of a = 1 in the solar system. Recently, the authors of [7] conduct further research on the orbital dynamics, such as equilibrium points and Hill region configurations, for the PN planar circular restricted Sun-Jupiter system by considering the motion equations derived in [6].…”

Section: Introductionmentioning

confidence: 99%

Non-truncated strategy to exactly integrate the post-Newtonian Lagrangian circular restricted three-body problem

2018

View full text Add to dashboard Cite

In this study, we present a novel non-truncated strategy by accompanying the fixed-point iteration with traditional numerical integrators. The proposed non-truncated strategy aims to exactly integrate implicit motion equations that are directly derived from the Lagrangian of the post-Newtonian circular restricted three-body problem. In comparison with the commonly used truncated approach, which cannot exactly but approximately preserve the generalized Jacobian constant (or energy) of the original Lagrangian system, the proposed non-truncated strategy has been determined to preserve this constant well. In fact, the non-truncated strategy and the truncated approach have a difference at second post-Newtonian order. Based on Kolmogorov-Arnold-Moser theory, this difference from the truncation in the equations of motion may lead to destroying the orbital configuration, dynamical behavior of order and chaos, and conservation of the post-Newtonian circular restricted three-body problem. The non-truncated strategy proposed in this study can avoid all these drawbacks and provide highly reliable and accurate numerical solutions for the post-Newtonian Lagrangian dynamics. Finally, numerical results show that the non-truncated strategy can preserve the generalized Jacobian constant in the accuracy of O(10 −12 ), whereas the truncated approach at the first post-Newtonian (1PN) order only has an accuracy of O(10 −3 ). Moreover, several orbits are observed to be escaping from the bounded region in the 1PN truncated system via the truncated strategy, but these escaping orbits are unobserved via the non-truncated strategy.

show abstract

On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem

Cited by 26 publications

References 30 publications

Equilibrium Points and Basins of Convergence in the Triangular Restricted Four-Body Problem with a Radiating Body

Equilibrium Points and Basins of Convergence in the Triangular Restricted Four-Body Problem with a Radiating Body

Unveiling the basins of convergence in the pseudo-Newtonian planar circular restricted four-body problem

Non-truncated strategy to exactly integrate the post-Newtonian Lagrangian circular restricted three-body problem

Contact Info

Product

Resources

About