Stability Analysis of the Rhomboidal Restricted Six-Body Problem

We discuss the restricted rhomboidal six-body problem (RR6BP), which has four positive masses at the vertices of the rhombus, and the fifth mass is at the intersection of the two diagonals. These masses always move in rhomboidal CC with diagonals 2 a and 2 b . The sixth body, having a very small mass, does not influence the motion of the five masses, also called primaries. The masses of the primaries are m 1 = m 2 = m 0 = m and m 3 = m 4 = m ˜ . The masses m and m ˜ are written as functions of parameters a and b such that they always form a rhomboidal central configuration. The evolution of zero velocity curves is discussed for fixed values of positive masses. Using the first integral of motion, we derive the region of possible motion of test particle m 5 and identify the value of Jacobian constant C for different energy intervals at which these regions become disconnected. Using semianalytical techniques, we show the existence and uniqueness of equilibrium solutions on the axes and off the axes. We show that, for b ∈ 1 / 3 , 1.1394282249562009 , there always exist 12 equilibrium points. We also show that all 12 equilibrium points are unstable.

Bifurcation Analysis and Periodic Solutions of the HD 191408 System with Triaxial and Radiative Perturbations

The nonlinear orbital dynamics of a class of the perturbed restricted three-body problem is studied. The two primaries considered here refer to the binary system HD 191408. The third particle moves under the gravity of the binary system, whose triaxial rate and radiation factor are also considered. Based on the dynamic governing equation of the third particle in the binary HD 191408 system, the motion state manifold is given. By plotting bifurcation diagrams of the system, the effects of various perturbation factors on the dynamic behavior of the third particle are discussed in detail. In addition, the relationship between the geometric configuration and the Jacobian constant is discussed by analyzing the zero-velocity surface and zero-velocity curve of the system. Then, using the Poincaré–Lindsted method and numerical simulation, the second- and third-order periodic orbits of the third particle around the collinear libration point in two- and three-dimensional spaces are analytically and numerically presented. This paper complements the results by Singh et al. [Singh et al., AMC, 2018]. It contains not only higher-order analytical periodic solutions in the vicinity of the collinear equilibrium points but also conducts extensive numerical research on the bifurcation of the binary system.

Heterogeneous primaries in CR4BP

This paper investigates the motion of the massless body moving under the influence of the gravitational forces of the three equal heterogeneous oblate spheroids placed at Lagrangian configuration. After determining the equations of motion and the Jacobian constant of the massless body, we have illustrated the numerical work (Stationary points, zero-velocity curves, regions of motion, Poincare surfaces of section and basins of attraction). And then we have checked the linear stability of the stationary points and found that all the stationary points are unstable.

Classification of Swing-By Trajectories Using the Moon

In the present paper we study and classify the swing-by maneuvers that use the Moon as the body for the close approach. The goal is to simulate a large variety of initial conditions for those orbits and classify them according to the effects caused by the close approach in the orbit of the spacecraft. Special attention is given to identify the regions where the captures and escapes occur. The classical three parameters (Jacobian constant, pericenter distance and angle of approach) used to identify a Swing-By maneuver are varied in large intervals to cover a large range of possibilities for the maneuver. Letter-plots figures are made to show the results obtained in a compact form. The theoretical prediction that for 0° ≤ ψ ≤ 180° the spacecraft losses energy and for 180° ≤ ψ ≤ 360° the spacecraft gains energy is confirmed. Regions containing trajectories that are candidates to generate Belbruno-Miller trajectories are identified. The well-known planar restricted circular three-body problem is used as the mathematical model. The equations are regularized (using Lamaiˆtre’s regularization), so it is possible to avoid the numerical problems that come from the close approach with the Moon.

The Cometary and Asteroidal Origins of Meteors

A quantitative examination of the gravitational and nongravitational changes of orbits shows that for larger interplanetary bodies the perturbations by Jupiter strongly predominate over all other effects, which include perturbations by other planets, splitting of comet nuclei and jet effects of cometary ejections. In an approximation to the restricted three-body problem, Sun-Jupiter-comet/asteroid, the value of the Jacobian integral represents a parameter of conspicuous stability which can be applied to delineate the evolutionary paths of the potential parent bodies of the meteoroids in the system of conventional orbital elements. Earth-crossing orbits can be reached along three main paths by the comets, and along two by the asteroids.The structure of meteor streams, however, indicates that the mutual compensation of the changes in individual elements entering the Jacobian integral, which is characteristic for the comets, does not work among the meteoroids. It appears that additional forces of a different kind must exert appreciable influence on the motion of interplanetary particles of meteoroid size. Nevertheless, the distribution of the Jacobian constant in various samples of meteor orbits, from those of faint Super-Schmidt meteors up to those of meteorite-dropping fireballs, furnishes some information on the type of their parent bodies and on the relative contribution of individual sources.