scholarly journals An Examination of the Mass Limit for Stability at the Triangular Lagrange Points for a Three-Body System and a Special Case of the Four-Body Problem

2015 ◽  
Author(s):  
Sean Kemp
Keyword(s):  
Body Problem ◽  
Body System ◽  
Mass Limit ◽  
Lagrange Points ◽  
Four Body Problem ◽  
Three Body ◽  
Special Case

scholarly journals Symmetric Four-mass Schubart-like Systems

2014 ◽  
Vol 9 (S310) ◽  
pp. 106-107 ◽  
Author(s):  
Winston L. Sweatman
Keyword(s):  
General Problem ◽  
Body Problem ◽  
Four Body Problem ◽  
Special Case ◽  
Simple Models

AbstractThe general four-body problem can be simplified by considering the special case where the system contains two pairs of identical masses and is symmetrical. The simple models that occur may aid our understanding of the general problem. Systems that arise from Schubart-like interplay orbits are an important feature of the dynamics.

scholarly journals Stability of the Sitnikov's circular restricted three body problem when the primaries are oblate spheroid

BIBECHANA ◽  
2014 ◽  
Vol 11 ◽  
pp. 149-156
Author(s):  
RR Thapa
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Oblate Spheroid ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Third Body ◽  
Independent Variable ◽  
The Stability ◽  
Three Body ◽  
Special Case

The Sitnikov's problem is a special case of restricted three body problem if the primaries are of equal masses (m1 = m2 = 1/2) moving in circular orbits under Newtonian force of attraction and the third body of mass m3 moves along the line perpendicular to plane of motion of primaries. Here oblate spheroid primaries are taken. The solution of the Sitnikov's circular restricted three body problem has been checked when the primaries are oblate spheroid. We observed that solution is depended on oblate parameter A of the primaries and independent variable τ = ηt. For this the stability of non-trivial solutions with the characteristic equation is studied. The general equation of motion of the infinitesimal mass under mutual gravitational field of two oblate primaries are seen at equilibrium points. Then the stability of infinitesimal third body m3 has been calculated. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10395 BIBECHANA 11(1) (2014) 149-156

On the Dynamical Properties of the Sun-Earth-Moon Restricted Four Body Problem

2013 ◽  
Vol 834-836 ◽  
pp. 1869-1872
Author(s):  
Yun Yan Zhang ◽  
Nan Li ◽  
Yan Jun Li
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Integral Relation ◽  
The Sun ◽  
Hill Region ◽  
Calculation Results ◽  
Four Body Problem ◽  
The Stability ◽  
Three Body ◽  
The Hill

The properties of the sun-earth-moon restricted four body problem were investigated. On the base of the earth-moon circular restricted three body problem, taking gravitation effect of the sun into account, a dynamic model of restricted four body problem was established. Equilibrium points were calculated by proper numerical method and linearization procedure was introduced to obtain the stability of these points. An invariant integral relation of this four body problem similar with Jacobi integral was derived, then the Hill region was investigated. Calculation results show that the equilibrium points will keep moving near libration points, also has their stability unchanged, and the Hill region will be non-periodically time varying.

DIFFERENT TYPES OF ATTRACTORS IN THE THREE BODY PROBLEM PERTURBED BY DISSIPATIVE TERMS

2011 ◽  
Vol 21 (08) ◽  
pp. 2195-2209 ◽  
Author(s):  
JOHN D. HADJIDEMETRIOU ◽  
GEORGE VOYATZIS
Keyword(s):  
Hamiltonian System ◽  
Degrees Of Freedom ◽  
Body Problem ◽  
Large Mass ◽  
Three Body Problem ◽  
Resonant Orbits ◽  
Different Types ◽  
Families Of Periodic Orbits ◽  
Three Body ◽  
Special Case

We study the evolution of a conservative dynamical system with three degrees of freedom, where small nonconservative terms are added. The conservative part is a Hamiltonian system, describing the motion of a planetary system consisting of a star, with a large mass, and of two planets, with small but not negligible masses, that interact gravitationally. This is a special case of the three body problem, which is nonintegrable. We show that the evolution of the system follows the topology of the conservative part. This topology is critically determined by the families of periodic orbits and their stability. The evolution of the complete system follows the families of the conservative part and is finally trapped in the resonant orbits of the Hamiltonian system, in different types of attractors: chaotic attractors, limit cycles or fixed points.

scholarly journals The Axisymmetric Central Configurations of the Four-Body Problem with Three Equal Masses

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 648
Author(s):  
Emese Kővári ◽  
Bálint Érdi
Keyword(s):  
Analytical Solution ◽  
Body Problem ◽  
Equal Mass ◽  
The Other ◽  
Coordinate Space ◽  
Four Body Problem ◽  
Axis Of Symmetry ◽  
The Masses ◽  
Special Case ◽  
Two Bodies

In the studied axisymmetric case of the central four-body problem, the axis of symmetry is defined by two unequal-mass bodies, while the other two bodies are situated symmetrically with respect to this axis and have equal masses. Here, we consider a special case of the problem and assume that three of the masses are equal. Using a recently found analytical solution of the general case, we formulate the equations of condition for three equal masses analytically and solve them numerically. A complete description of the problem is given by providing both the coordinates and masses of the bodies. We show furthermore how the three-equal-mass solutions are related to the general case in the coordinate space. The physical aspects of the configurations are also studied and discussed.

scholarly journals Properties of the lunar gravity assisted transfers from LEO to the retrograde-GEO

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bo-yong He ◽  
Peng-bin Ma ◽  
Heng-nian Li
Keyword(s):  
Body Problem ◽  
Space Debris ◽  
Low Earth Orbit ◽  
Earth Orbit ◽  
Restricted Three Body Problem ◽  
The Sun ◽  
Lunar Gravity ◽  
Three Body Problem ◽  
Four Body Problem ◽  
Three Body

AbstractThe retrograde geostationary earth orbit (retro-GEO) is an Earth’s orbit. It has almost the same orbital altitude with that of a GEO, but an inclination of 180°. A retro-GEO monitor-satellite gives the GEO-assets vicinity space-debris warnings per 12 h. For various reasons, the westward launch direction is not compatible or economical. Thereby the transfer from a low earth orbit (LEO) to the retro-GEO via once lunar swing-by is a priority. The monitor-satellite departures from LEO and inserts into the retro-GEO both using only one tangential maneuver, in this paper, its transfer’s property is investigated. The existence of this transfer is verified firstly in the planar circular restricted three-body problem (CR3BP) model based on the Poincaré-section methodology. Then, the two-impulse values and the perilune altitudes are computed with different transfer durations in the planar CR3BP. Their dispersions are compared with different Sun azimuths in the planar bi-circular restricted four-body problem (BR4BP) model. Besides, the transfer’s inclination changeable capacity via lunar swing-by and the Sun-perturbed inclination changeable capacity are investigated. The results show that the two-impulse fuel-optimal transfer has the duration of 1.76 TU (i.e., 7.65 days) with the minimum values of 4.251 km s−1 in planar CR3BP, this value has a range of 4.249–4.252 km s−1 due to different Sun azimuths in planar BR4BP. Its perilune altitude changes from 552.6 to 621.9 km. In the spatial CR3BP, if the transfer duration is more than or equal to 4.00 TU (i.e., 17.59 days), the lunar gravity assisted transfer could insert the retro-GEO with any inclination. In the spatial BR4BP, the Sun’s perturbation does not affect this conclusion in most cases.

STABILIZATION OF CHAOTIC BEHAVIOR IN THE RESTRICTED THREE-BODY PROBLEM

2007 ◽  
Vol 17 (10) ◽  
pp. 3603-3606 ◽  
Author(s):  
ARSEN DZHANOEV ◽  
ALEXANDER LOSKUTOV
Keyword(s):  
Body Problem ◽  
Chaotic Behavior ◽  
Melnikov Method ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lagrange Points ◽  
Three Body ◽  
Two Bodies

The restricted three-body problem on the example of a perturbed Sitnikov case is considered. On the basis of the Melnikov method we study a possibility to stabilize the obtained chaotic solutions by two bodies placed in the triangular Lagrange points. It is shown that in this case, in addition to regular and chaotic solutions, there exist stabilized solutions.

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