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1.
The restricted (equilateral) four-body problem consists of three bodies of masses m 1, m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them. 相似文献
2.
In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented.
Three bodies of masses m
1, m
2 and m
3 (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the
system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction
of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families
and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal
and finally when we have three unequal masses. Series, with respect to the mass m
3, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of
the mass parameters are also calculated. 相似文献
3.
Special analytical solutions are determined for restricted, coplanar, four-body equal mass problems, including the Caledonian problem, where the masses Mi = M for i = 1,2,3,4. Most of these solutions are shown to reduce to the Lagrange solutions of the Copenhagen problem of three bodies by reducing two of the masses (mi = m for i = 1,2) in the four-body equal mass problem to zero while maintaining their equality of mass. In so doing, families of special solutions to the four-body problem are shown to exist for any value of the mass ratio μ = m/M. 相似文献
4.
We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses.
Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two
mass ratios as μ
1 = m
1/M
T and μ
2 = m
2/M
T, where m
1 and m
2 are the two unequal masses and M
T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four
types of four-body problems. 相似文献
5.
Lennard F. Bakker Tiancheng Ouyang Duokui Yan Skyler Simmons 《Celestial Mechanics and Dynamical Astronomy》2011,110(3):271-290
We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise
symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric
periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. We then
use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar
pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability
when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53. 相似文献
6.
We study numerically the photogravitational version of the problem of four bodies, where an infinitesimal particle is moving under the Newtonian gravitational attraction of three bodies 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. The fourth body does not affect the motion of the three bodies (primaries). We consider that the primary body m 1 is dominant and is a source of radiation while the other two small primaries m 2 and m 3 are equal. In this case (photogravitational) we examine the linear stability of the Lagrange triangle solution. 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 on the orbital plane are given. The existence and the number of the collinear and the non-collinear equilibrium points of the problem depends on the mass parameters of the primaries and the radiation factor q 1. Critical masses m 3 and radiation q 1 associated with the existence and the number of the equilibrium points are given. The stability of the relative equilibrium solutions in all cases are also studied. In the last section we investigate the existence and location of the out of orbital plane equilibrium points of the problem. We found that such critical points exist. These points lie in the (x,z) plane in symmetrical positions with respect to (x,y) plane. The stability of these points are also examined. 相似文献
7.
We have two mass points of equal masses m
1=m
2 > 0 moving under Newton’s law of attraction in a non-collision parabolic orbit while their center of mass is at rest. We
consider a third mass point, of mass m
3=0, moving on the straight line L perpendicular to the plane of motion of the first two mass points and passing through their
center of mass. Since m
3=0, the motion of m
1 and m
2 is not affected by the third and from the symmetry of the motion it is clear that m
3 will remain on the line L. The parabolic restricted three-body problem describes the motion of m
3. Our main result is the characterization of the global flow of this problem. 相似文献
8.
In this paper, we extend the basic model of the restricted four-body problem introducing two bigger dominant primaries m 1 and m 2 as oblate spheroids when masses of the two primary bodies (m 2 and m 3) are equal. The aim of this study is to investigate the use of zero velocity surfaces and the Poincaré surfaces of section to determine the possible allowed boundary regions and the stability orbit of the equilibrium points. According to different values of Jacobi constant C, we can determine boundary region where the particle can move in possible permitted zones. The stability regions of the equilibrium points expanded due to presence of oblateness coefficient and various values of C, whereas for certain range of t (100≤t≤200), orbits form a shape of cote’s spiral. For different values of oblateness parameters A 1 (0<A 1<1) and A 2 (0<A 2<1), we obtain two collinear and six non-collinear equilibrium points. The non-collinear equilibrium points are stable when the mass parameter μ lies in the interval (0.0190637,0.647603). However, basins of attraction are constructed with the help of Newton Raphson method to demonstrate the convergence as well as divergence region of the equilibrium points. The nature of basins of attraction of the equilibrium points are less effected in presence and absence of oblateness coefficients A 1 and A 2 respectively in the proposed model. 相似文献
9.
Rogerio Deienno Diogo Merguizo Sanchez Antonio Fernando Bertachini de Almeida Prado Georgi Smirnov 《Celestial Mechanics and Dynamical Astronomy》2016,124(4):433-456
We analyze the families of central configurations of the spatial 5-body problem with four masses equal to 1 when the fifth mass m varies from 0 to \(+\infty \). In particular we continue numerically, taking m as a parameter, the central configurations (which all are symmetric) of the restricted spatial (\(4+1\))-body problem with four equal masses and \(m=0\) to the spatial 5-body problem with equal masses (i.e. \(m=1\)), and viceversa we continue the symmetric central configurations of the spatial 5-body problem with five equal masses to the restricted (\(4+1\))-body problem with four equal masses. Additionally we continue numerically the symmetric central configurations of the spatial 5-body problem with four equal masses starting with \(m=1\) and ending in \(m=+\infty \), improving the results of Alvarez-Ramírez et al. (Discrete Contin Dyn Syst Ser S 1: 505–518, 2008). We find four bifurcation values of m where the number of central configuration changes. We note that the central configurations of all continued families varying m from 0 to \(+\infty \) are symmetric. 相似文献
10.
Montserrat Corbera Josep Maria Cors Jaume Llibre 《Celestial Mechanics and Dynamical Astronomy》2011,109(1):27-43
We consider the Newtonian four-body problem in the plane with a dominat mass M. We study the planar central configurations of this problem when the remaining masses are infinitesimal. We obtain two different
classes of central configurations depending on the mutual distances between the infinitesimal masses. Both classes exhibit
symmetric and non-symmetric configurations. And when two infinitesimal masses are equal, with the help of extended precision
arithmetics, we provide evidence that the number of central configurations varies from five to seven. 相似文献
11.
Marcelo Marchesin Deborah da Paixão Vasconcellos 《Astrophysics and Space Science》2014,352(2):443-460
We study a highly symmetric nine-body problem in which eight positive masses, called the primaries, move four by four, in two concentric circular motions such that their configuration is always a square for each group of four masses. The ninth body being of negligible mass and not influencing the motion of the eight primaries. We assume all the nine masses are in the same plane and that the masses of the primaries are \(m_{1}=m_{2}=m_{3}=m_{4}=\tilde{m}\) and m 5=m 6=m 7=m 8=m and the radii associated to the circular motion of the bodies with mass \(\tilde{m}\) is λ∈[λ 0,1] and for the bodies with mass m is 1. We prove the existence of central configurations which characterize such arrangement of the primaries and we study the influence of the parameter λ, the ratio of the radii of the two circles, on the masses m and \(\tilde{m}\) . We use a synodical system of coordinates to eliminate the time dependence on the equations of motion. We show the existence of equilibria solutions symmetrically distributed on the four quadrants and their dependence on the parameter λ. Finally, we show that there can be 13, 17 or 25 equilibria solutions depending on the size of λ and we investigate their linear stability. 相似文献
12.
This article deals with the region of motion in the Sitnikov four-body problem where three bodies (called primaries) of equal masses fixed at the vertices of an equilateral triangle. Fourth mass which is finite confined to moves only along a line perpendicular to the instantaneous plane of the motions of the primaries. Contrary to the Sitnikov problem with one massless body the primaries are moving in non-Keplerian orbits about their centre of mass. It is investigated that for very small range of energy h the motion is possible only in small region of phase space. Condition of bounded motions has been derived. We have explored the structure of phase space with the help of properly chosen surfaces of section. Poincarè surfaces of section for the energy range ?0.480≤h≤?0.345 have been computed. We have chosen the plane (q 1,p 1) as surface of section, with q 1 is the distance of a primary from the centre of mass. We plot the respective points when the fourth body crosses the plane q 2=0. For low energy the central fixed point is stable but for higher value of energy splits in to an unstable and two stable fixed points. The central unstable fixed point once again splits for higher energy into a stable and three unstable fixed points. It is found that at h=?0.345 the whole phase space is filled with chaotic orbits. 相似文献
13.
The collinear equilibrium position of the circular restricted problem with the two primaries at unit distance and the massless body at the pointL 3 is extended to the planar three-body problem with respect to the massm 3 of the third body; the mass ratio μ of the two primaries is considered constant and the constant angular velocity of the straight line on which the three masses stay at rest is taken equal to 1. As regards periodic motions ‘around’ the equilibrium pointL 3, four possible extensions from the restricted to the general problem are presented each of them starting with a simple or a doubly periodic orbit of the family α of the Copenhagen category (μ=0.50). Form 3=0.10, μ=0.50 (i.e. for fixed masses of all three bodies) the characteristic curve of the extended family α is found. The qualitative differences of the families corresponding tom 3=0 andm 3=0.10 are discussed. 相似文献
14.
Gregory Buck 《Celestial Mechanics and Dynamical Astronomy》1991,51(4):305-317
Moulton's Theorem says that given an ordering of masses m 1, ..., m nthere exists a unique collinear central configuration. The theorem allows us to ask the questions: What is the distribution of n equal masses in the collinear central configuration? What is the behavior of the distribution as n → ∞? These questions are due to R. Moekel (personal conversation). Central configurations are found to be attracting fixed points of a flow — a flow we might call an auxiliary flow (in the text it is denoted F(X)), since it has little to do with the equations of motion. This flow is studied in an effort to characterize the mass distribution. Specifically, for a collinear central configuration of n equal masses, a bound is found for the position of the masses furthest from the center of mass. Also some facts concerning the distribution of the inner masses are discovered. 相似文献
15.
We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m
1 = m
2 = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m
3 < 10−3, placed initially on the z-axis. We begin by finding for the restricted problem (with m
3 = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m
3 increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of
saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m
3 ≈ 10−6, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m
3 = 0. Furthermore, as m
3 increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m
3 values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries,
as observed in the m
3 = 0 case. 相似文献
16.
P. S. Soulis K. E. Papadakis T. Bountis 《Celestial Mechanics and Dynamical Astronomy》2008,100(4):251-266
We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits
in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the
fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted
three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż0 varies within a finite interval (while z
0 tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve
corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D
branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these
orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the
z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion
near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of
ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity. 相似文献
17.
Lennard F. Bakker Tiancheng Ouyang Duokui Yan Skyler Simmons Gareth E. Roberts 《Celestial Mechanics and Dynamical Astronomy》2010,108(2):147-164
We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous
binary collision orbits in the symmetric collinear four-body problem with masses 1, m, m, 1, and also in a symmetric planar
four-body problem with equal masses. In both problems, the assumed symmetries reduce the determination of linear stability
to the numerical computation of a single real number. For the collinear problem, this verifies the earlier numerical results
of Sweatman for linear stability with respect to collinear and symmetric perturbations. 相似文献
18.
We present some results of a numerical exploration of the rectilinear problem of three bodies, with the two outer masses equal. The equations of motion are first given in relative coordinates and in regularized variables, removing both binary collision singularities in a single coordinate transformation. Among our most important results are seven periodic solutions and three symmetric triple collision solutions. Two of these periodic solutions have been continued into families, the outer massm
3 being the family parameter. One of these families exists for all masses while the second family is a branch of the first at a second-kind critical orbit. This last family ends in a triple collision orbit.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978. 相似文献
19.
20.
Based on visual estimates by AAVSO observers, we have constructed light curves for 80 Galactic novae flared up in 1986–2006 and determined the photometric parameters m vis(max), t 2, and t 3 for 64 novae. Using the empirical relation M V (max) = ?10.66(±0.33) + 2.31(±0.26) × log t 2, we have obtained the absolute magnitudes at maximum and apparent distance moduli of the novae. 相似文献