Exploring the Location and Linear Stability of the Equilibrium Points in the Equilateral Restricted Four-Body Problem

The planar version of the equilateral restricted four-body problem, with three unequal masses, is numerically investigated. By adopting the grid classification method we locate the coordinates, on the plane [Formula: see text], of the points of equilibrium, for all possible values of the masses of the primaries. The linear stability of the libration points is also determined, as a function of the masses. Our analysis indicates that linearly stable points of equilibrium exist only when one of the primaries has a considerably larger mass, with respect to the other two primary bodies, when the triangular configuration of the primaries is also dynamically stable.

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Equilibrium Points and Basins of Convergence in the Triangular Restricted Four-Body Problem with a Radiating Body

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420300037 ◽

2020 ◽

Vol 30 (02) ◽

pp. 2030003 ◽

Cited By ~ 3

Author(s):

J. E. Osorio-Vargas ◽

Guillermo A. González ◽

F. L. Dubeibe

Keyword(s):

Body Problem ◽

Equilibrium Points ◽

Center Of Mass ◽

Libration Points ◽

The Sun ◽

Radiation Factor ◽

Trojan Asteroid ◽

Four Body Problem ◽

Common Center ◽

The Stability

In this paper, we extend the basic equilateral four-body problem by introducing the effect of radiation pressure, Poynting–Robertson drag, and solar wind drag. In our setup, three primaries lie at the vertices of an equilateral triangle and move in circular orbits around their common center of mass. Here, one of the primaries is a radiating body and the fourth body (whose mass is negligible) does not affect the motion of the primaries. We show that the existence and the number of equilibrium points of the problem depend on the mass parameters and radiation factor. Consequently, the allowed regions of motion, the regions of the basins of convergence for the equilibrium points, and the basin entropy will also depend on these parameters. The present dynamical model is analyzed for three combinations of mass for the primaries: equal masses, two equal masses, different masses. As the main results, we find that in all cases the libration points are unstable if the radiation factor is larger than 0.01 and hence able to destroy the stability of the libration points in the restricted four-body problem composed by the Sun, Jupiter, Trojan asteroid and a test (dust) particle. Also, we conclude that the number of fixed points decreases with the increase of the radiation factor.

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Symmetry of planar four-body convex central configurations

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.0320 ◽

2008 ◽

Vol 464 (2093) ◽

pp. 1355-1365 ◽

Cited By ~ 45

Author(s):

Alain Albouy ◽

Yanning Fu ◽

Shanzhong Sun

Keyword(s):

Body Problem ◽

The Other ◽

Central Configurations ◽

Central Configuration ◽

Geometric Properties ◽

Four Body Problem ◽

Planar Central Configurations ◽

The Masses ◽

The Relationship ◽

Convex Central Configuration

We study the relationship between the masses and the geometric properties of central configurations. We prove that, in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.

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

Symmetry ◽

10.3390/sym12040648 ◽

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.

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