Approximate Analytical Periodic Solutions to the Restricted Three-Body Problem with Perturbation, Oblateness, Radiation and Varying Mass

Against the background of a restricted three-body problem consisting of a supergiant eclipsing binary system, the two primaries are composed of a pair of bright oblate stars whose mass changes with time. The zero-velocity surface and curve of the problem are numerically studied to describe the third body’s motion area, and the corresponding five libration points are obtained. Moreover, the effect of small perturbations, Coriolis and centrifugal forces, radiative pressure, and the oblateness and mass parameters of the two primaries on the third body’s dynamic behavior is discussed through the bifurcation diagram. Furthermore, the second- and third-order approximate analytical periodic solutions around the collinear solution point L3 in two-dimensional plane and three-dimensional spaces are presented by using the Lindstedt-Poincaré perturbation method.

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.

Minimal velocity surface in the restricted circular Three-Body-Problem

In the framework of the restricted circular Three-Body-Problem, the concept of the minimum velocity surface S is introduced, which is a modification of the zero-velocity surface (Hill surface). The existence of Hill surface requires occurrence of the Jacobi integral. The minimum velocity surface, other than the Jacobi integral, requires conservation of the sector velocity of a zero-mass body in the projection on the plane of the main bodies motion. In other words, there must exist one of the three angular momentum integrals. It is shown that this integral exists for a dynamic system obtained after a single averaging of the original system by longitude of the main bodies. Properties of S are investigated. Here is the most significant. The set of possible motions of the zero-mass body bounded by the surface S is compact. As an example the surfaces S for four small moons of Pluto are considered in the framework of the averaged problem Pluto — Charon — small satellite. In all four cases, S represents a topological torus with small cross section, having a circumference in the plane of motion of the main bodies as the center line.

Equilibria and orbits in the dynamical environment of asteroid 22 Kalliope

This paper studies the orbital dynamics of the potential of asteroid 22 Kalliope using observational data of the irregular shape. The zero-velocity surface are calculated and showed with different Jacobian values. All five equilibrium points are found, four of them are outside and unstable, and the other one is inside and linearly stable. The movement and bifurcations of equilibrium points during the variety of rotation speed and density of the body are investigated. The Hopf bifurcations occurs during the variety of rotational speed from ω=1.0ω0 to 0.5ω0, and the Saddle-Node bifurcation occurs during the variety of rotational speed from ω=1.0ω0 to 2.0ω0. Both unstable and stable resonant periodic orbits around Kalliope are coexisting. The perturbation of an unstable periodic orbit shows that the gravitational field of Kalliope is strongly perturbed.

Numerical Study of the Zero Velocity Surface and Transfer Trajectory of a Circular Restricted Five-Body Problem

We focus on a type of circular restricted five-body problem in which four primaries with equal masses form a regular tetrahedron configuration and circulate uniformly around the center of mass of the system. The fifth particle, which can be regarded as a small celestial body or probe, obeys the law of gravity determined by the four primaries. The geometric configuration of zero-velocity surfaces of the fifth particle in the three-dimensional space is numerically simulated and addressed. Furthermore, a transfer trajectory of the fifth particle skimming over four primaries then is designed.

Dwarfs in the entourage of the Local Volume groups: flow tracers and cosmological probes

We consider a sample of dwarf galaxies with accurate distances and velocities around 14 massive groups in the Local Volume. We combine all the data into a single synthetic group, and then determine its radius of the zero-velocity surface, separating it against the global cosmic expansion. Our estimation is derived from fitting the the spherical infall model (including effects of the cosmological constant) to the observational data.We found the optimal value of the radius to be 0.93 ± 0.02 Mpc. Assuming the Planck model parameters, it corresponds to the total mass of the synthetic group (1.6 ± 0.2) × 1012M⊙. Thus, we obtain the paradoxical result that the total mass of the synthetic group estimated on the scale of 3–4 its virial radius is only 60% of the virial mass estimate. Anyway, we conclude that wide outskirts of the nearby groups do not contain a large amount of hidden mass outside their virial radii.

Cosmic flow around local massive galaxies

Aims. We use accurate data on distances and radial velocities of galaxies around the Local Group, as well as around 14 other massive nearby groups, to estimate their radius of the zero-velocity surface, R0, which separates any group against the global cosmic expansion. Methods. Our R0 estimate was based on fitting the data to the velocity field expected from the spherical infall model, including effects of the cosmological constant. The reported uncertainties were derived by a Monte Carlo simulation. Results. Testing various assumptions about a location of the group barycentre, we found the optimal estimates of the radius to be 0.91 ± 0.05 Mpc for the Local Group, and 0.93 ± 0.02 Mpc for a synthetic group stacked from 14 other groups in the Local Volume. Under the standard Planck model parameters, these quantities correspond to the total mass of the group ~ (1.6 ± 0.2) × 1012M⊙. Thus, we are faced with the paradoxical result that the total mass estimate on the scale of R0 ≈ (3−4)Rvir is only 60% of the virial mass estimate. Anyway, we conclude that wide outskirts of the nearby groups do not contain a large amount of hidden mass outside their virial radius.

EQUILIBRIUM POINTS AND THEIR STABILITY IN THE RESTRICTED FOUR-BODY PROBLEM

We study numerically the problem of four bodies, three of which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle, while the fourth is infinitesimal. The fourth body does not affect the motion of the three bodies (primaries). The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves as well as the positions of the equilibrium points are given. The existence and the number of collinear and noncollinear equilibrium points of the problem depend on the mass parameters of the primaries. For three unequal masses, collinear equilibrium solutions do not exist. Critical masses associated with the existence and the number of equilibrium points, are given. The stability of the relative equilibrium solutions in all cases is also studied. The regions of the basins of attraction for the equilibrium points of the present dynamical model for some values of the mass parameters are illustrated.