A model for binary-binary close encounters and collisions from a dynamical point of view

Medina²,

Vidal

2015

Astrophys Space Sci

Self Cite

This paper consider the trapezoidal collinear fourbody problem as a model for binary-binary gravitational interaction in star clusters. It has two degrees of freedom, it also is a sub-problem of the trapezoidal four-body problem that has three degrees of freedom. We prove that all its singularities in the trapezoidal four-body problem are due to collision and give some results regarding escapes to infinity. In order to give a description of the orbits suffering collisions between two bodies while the other two escape to infinity with zero radial velocity (parabolic orbits), we make use of the McGehee's techniques to blow-up-regularize the singularities due to collision and infinity.

Section: Introductionmentioning

confidence: 99%

The trapezoidal collinear four-body problem

Medina²,

Vidal

2015

Astrophys Space Sci

Self Cite

“…In this section, we give the conditions for the undergoing potential, U in (1), recall some particular examples, derive the regularized equations of the model and present the main features (we consider a suitable Poincaré section and define the collision manifold) that will play an essential role along the paper.…”

Section: General Settingmentioning

confidence: 99%

“…One direction to tackle this problem has been to consider few body problems, as the collinear three body problem [15,7] or the isosceles three body problem [6]. Some others have some symmetries involved, as the symmetric collinear four body problem [10,1,4], the trapezoidal four body problem [9,3,2], and the rhomboidal four body problem [5] and [12]. Some of them depend on parameters associated to the masses.…”

Section: Introductionmentioning

confidence: 99%

Ejection–Collision Orbits in Two Degrees of Freedom Problems in Celestial Mechanics

Barrabés

Medina³

et al. 2021

J Nonlinear Sci

In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of N -body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types of ejection-collision orbits, that is, orbits that start and end at total collision. Such orbits are regarded as heteroclinic connections between two equilibrium points and are mainly characterized by the partial collisions that the trajectories find on their way. The proof of their existence is based on the transversality of 2D-invariant manifolds and on the behavior of the dynamics on the total collision manifold, both of them are thoroughly described.

“…At such a time the potential energy approaches infinity, the equations of motion become undefined and the solution has a singularity. The analytical and numerical study of this problem requires the McGehee's blow up technique to regularize the singularity corresponding to total (quadruple) collision and the regularization of binary collisions (collisions between m 1 and m 2 ) and simultaneous binary collisions (m 1 and m 3 collide as well as m 2 and m 4 ), see for example [9] and [10]. This singularity due to total collision is blown up and in its place is glued an invariant total collision manifold.…”

Section: Introductionmentioning

confidence: 99%

“…Later on, Ouyang and Yan [15] and Huang [16] analytically prove the existence of such Schubart periodic orbits by applying topological methods and variational calculus, respectively. We finally mention, for the symmetric collinear four-body problem, the papers by Alvarez et al [9,17], where the authors provide some analytical results concerning singularities and regularization, and analytically study the quadruple collision manifold, the equilibrium points, the infinity manifold and the relation between both manifolds which allow them to prove the existence of orbits connecting quadruple collision and infinity. For the collinear non sym-metric four-body problem, in [18], Mather and McGehee prove the existence of solutions which become unbounded in finite time for special values of the masses.…”

Section: Introductionmentioning

confidence: 99%

Ejection-Collision orbits in the symmetric collinear four–body problem

Communications in Nonlinear Science and Numerical Simulation

Barrabés

Medina³

et al. 2019

In this paper, we consider the collinear symmetric four-body problem, where four masses m 3 = α, m 1 = 1, m 2 = 1, and m 4 = α, α > 0, are aligned in this order and move symmetrically about their center of mass. We introduce regularized variables to deal with binary collisions as well as McGehee coordinates to study the quadruple collision manifold for a negative value of the energy. The paper is mainly focused on orbits that eject from (or collide to) quadruple collision. The problem has two hyperbolic equilibrium points, located in the quadruple collision manifold. We use high order parametrizations of their stable/unstable manifolds to devise a numerical procedure to compute ejectioncollision orbits, for any value of α. Some results from the explorations done for α = 1 are presented. Furthermore, we prove the existence of ejection-direct escape orbits, which perform a unique type of binary collisions.