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.

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

2020 ◽  
Vol 30 (10) ◽  
pp. 2050155
Author(s):  
Euaggelos E. Zotos
Keyword(s):  
Linear Stability ◽  
Body Problem ◽  
Equilibrium Points ◽  
Libration Points ◽  
The Other ◽  
Classification Method ◽  
Four Body Problem ◽  
The Masses ◽  
Stable Points ◽  
Planar Version

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.

scholarly journals Symmetry of planar four-body convex central configurations

2008 ◽  
Vol 464 (2093) ◽  
pp. 1355-1365 ◽  
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.

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 A self-consistent weak friction model for the tidal evolution of circumbinary planets

2019 ◽  
Vol 627 ◽  
pp. A109 ◽  
Author(s):  
F. A. Zoppetti ◽  
C. Beaugé ◽  
A. M. Leiva ◽  
H. Folonier
Keyword(s):  
Body Problem ◽  
Semimajor Axis ◽  
Orbital Evolution ◽  
Friction Model ◽  
The Other ◽  
Tidal Evolution ◽  
Other Hand ◽  
Self Consistent ◽  
The Masses ◽  
Circumbinary Planets

We present a self-consistent model for the tidal evolution of circumbinary planets that is easily extensible to any other three-body problem. Based on the weak-friction model, we derive expressions of the resulting forces and torques considering complete tidal interactions between all the bodies of the system. Although the tidal deformation suffered by each extended mass must take into account the combined gravitational effects of the other two bodies, the only tidal forces that have a net effect on the dynamic are those that are applied on the same body that exerts the deformation, as long as no mean-motion resonance exists between the masses. As a working example, we applied the model to the Kepler-38 binary system. The evolution of the spin equations shows that the planet reaches a stationary solution much faster than the stars, and the equilibrium spin frequency is sub-synchronous. The binary components, on the other hand, evolve on a longer timescale, reaching a super-synchronous solution very close to that derived for the two-body problem. The orbital evolution is more complex. After reaching spin stationarity, the eccentricity was damped in all bodies and for all the parameters analysed here. A similar effect is noted for the binary separation. The semimajor axis of the planet, on the other hand, may migrate inwards or outwards, depending on the masses and orbital parameters. In some cases the secular evolution of the system may also exhibit an alignment of the pericenters, requiring the inclusion of additional terms in the tidal model. Finally, we derived analytical expressions for the variational equations of the orbital evolution and spin rates based on low-order elliptical expansions in the semimajor axis ratioαand the eccentricities. These are found to reduce to the well-known two-body case whenα→ 0 or when one of the masses is taken as equal to zero. This model allows us to find a closed and simple analytical expression for the stationary spin rates of all the bodies, as well as predicting the direction and magnitude of the orbital migration.

The Existence of Retrograde Orbits for the Four-Body Problem with Various Choices of Masses

2013 ◽  
Vol 871 ◽  
pp. 101-106
Author(s):  
Chong Li
Keyword(s):  
Variational Methods ◽  
Periodic Orbits ◽  
Opposite Direction ◽  
Body Problem ◽  
Topological Type ◽  
The Other ◽  
Action Functional ◽  
Four Body Problem

In this paper, we study the planar Newtonian four-body problem with various choices of masses. We prove that there exist infinitely many periodic and quasi-periodic orbits with certain topological type, called retrograde orbits, that minimize the action functional on certain path spaces. On these orbits, two particles revolve around each other in one direction, while the other two particles travel on themselves orbits in opposite direction, respectively. Our proof is based on variational methods inspired by the work of Kuo-Chang Chen.

scholarly journals New Phenomena in the Spatial Isosceles Three-Body Problem

2015 ◽  
Vol 25 (09) ◽  
pp. 1550116 ◽  
Author(s):  
Duokui Yan ◽  
Tiancheng Ouyang
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
The Other ◽  
Initial Condition ◽  
Collision Orbit ◽  
Three Body Problem ◽  
Straight Line ◽  
Three Body ◽  
New Phenomena ◽  
Two Bodies

In the three-body problem, it is known that there exists a special set of periodic orbits: spatial isosceles periodic orbits. In each period, one body moves up and down along a straight line, and the other two bodies rotate around this line. In this work, we revisit this set of orbits by applying variational method. Two unexpected phenomena are discovered. First, this set is not always spatial. It actually bifurcates from the circular Euler (central configuration) orbit to the Broucke (collision) orbit. Second, one of the orbits in this set encounters an oscillating behavior. By running its initial condition, the orbit stays periodic for only a few periods before it becomes irregular. However, it moves close to another periodic shape in a while. Shortly it falls apart again and starts running close to a third periodic shape after a moment. This oscillation continues as t increases. Actually, up to t = 1.2 × 105, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions.

scholarly journals Dynamical Aspects of an Equilateral Restricted Four-Body Problem

2009 ◽  
Vol 2009 ◽  
pp. 1-23 ◽  
Author(s):  
Martha Álvarez-Ramírez ◽  
Claudio Vidal
Keyword(s):  
Degrees Of Freedom ◽  
Body Problem ◽  
Center Of Mass ◽  
Hamiltonian Function ◽  
First Integral ◽  
Equilibrium Solutions ◽  
Integral Of Motion ◽  
Planar Case ◽  
Four Body Problem ◽  
The Masses

The spatial equilateral restricted four-body problem (ERFBP) is a four body problem where a mass point of negligible mass is moving under the Newtonian gravitational attraction of three positive masses (called the primaries) which move on circular periodic orbits around their center of mass fixed at the origin of the coordinate system such that their configuration is always an equilateral triangle. Since fourth mass is small, it does not affect the motion of the three primaries. In our model we assume that the two masses of the primariesm2andm3are equal toμand the massm1is1−2μ. The Hamiltonian function that governs the motion of the fourth mass is derived and it has three degrees of freedom depending periodically on time. Using a synodical system, we fixed the primaries in order to eliminate the time dependence. Similarly to the circular restricted three-body problem, we obtain a first integral of motion. With the help of the Hamiltonian structure, we characterize the region of the possible motions and the surface of fixed level in the spatial as well as in the planar case. Among other things, we verify that the number of equilibrium solutions depends upon the masses, also we show the existence of periodic solutions by different methods in the planar case.

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