On the dynamics of planar oscillations for a dumbbell satellite in $$\varvec{J_{2}}$$ J 2 problem

Abstract

In the present paper, we study regular and chaotic dynamics from planar oscillations of a dumbbell satellite under the influence of the gravity field generated by an oblate body, considering the effect of the zonal harmonic parameter J 2 . We theoretically show the existence of chaotic oscillations provided that the eccentricity becomes arbitrarily small, and the parameter J 2 is of the same order of magnitude as the eccentricity. This is carried out by applying the so-called Melnikov method. Finally, for arbitrarily chosen values for the parameters involved in such a problem, we study the transition from regular to chaotic oscillations for a dumbbell satellite via the analysis of chaotic maps and Poincaré surfaces of section, respectively.

Chaotic Dynamics in the Planar Gravitational Many-Body Problem with Rigid Body Rotations

“…The approach has also been used in the study of nonlinear PDEs, such as a reduction of the perturbed KdV equation Grimshaw & Tian (1994). Of immediate relevance to our problem, in addition to the recent application to the dumbbell satellite problem Fernández-Martínez et al (2016), the method has been applied to restricted three-body problems Xia (1992).…”

“…To do so, Lages et al (2017) generalized the Kepler map technique Meiss (1992) to describe the motion of a particle in the gravitational field of a such a rotating irregular body modeled by a dumbbell. Regarding the rotations of such an irregular object itself, Fernández-Martínez et al (2016) recently studied dynamics from planar oscillations of a dumbbell satellite orbiting an oblate body of much larger mass, obtaining a kind of generalized Beletsky equation. They show the existence of chaotic orbits in their system via the Melnikov method, and study the transition between regular to chaotic orbits numerically through the use of Poincaré maps.…”

Periodic Orbits of Third Kind in the Zonal J2 + J3 Problem

“…For more details about the zonal problem, see the classical work of [9]. Other recent works where the J 2 problem and the zonal RTBP were studied with a non-averaging approach are [10][11][12] respectively.…”

Periodic solutions for a dumbbell satellite equation

“…The aim of this paper is to study the periodic oscillations of a dumbbell satellite whose center of mass describes, under the gravity field generated by an oblate body, an elliptical oscillation of eccentricity e ∈ [0, 1), taking into account the effect of the zonal harmonic parameter J 2 . The planar oscillations of such dumbbell satellite systems can be described as the following second-order differential equation, first appeared in [2,11]:…”

“…which gives the classical Beletsky equation [5]. In a recent paper [11], based on the Melnikov method, the regular and chaotic dynamics from planar oscillations of a dumbbell satellite which can be described as Eq. (1.1) have been studied.…”