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1.
In this paper, we study the existence of transversal homoclinic orbits in a planar circular restricted four-body problem, based on the perturbation theory of integrable Hamiltonian systems. We start from a planar circular restricted four-body model and regard it as a perturbation of the two-body model. Then, in order to conveniently study unbounded orbits, we transform the infinite points to finite points by a non-canonical transformation, arriving at a non-Hamiltonian system with degenerate fixed points. According to the extended Melnikov method, we finally prove that there exist transversal homoclinic orbits in this four-body model. 相似文献
2.
In this paper we study the existence of a Smale horseshoe in a planar circular restricted four-body problem. For this planar four-body system there exists a transversal homoclinic orbit, but the fixed point is a degenerate saddle, so that the standard Smale–Birkhoff homoclinic theorem cannot be directly applied. We therefore apply the Conley–Moser conditions to prove the existence of a Smale horseshoe. Specifically, we first use the transversal structure of stable and unstable manifolds to make a linear transformation and then introduce a nonlinear Poincaré map $P$ by considering the truncated flow near the degenerate saddle; based on this Poincaré map $P$ , we define an invertible map $f$ , which is a composite function; by carefully checking the satisfiability of the Conley–Moser conditions for $f$ we finally prove that $f$ is a Smale horseshoe map, which implies that our restricted four-body problem has the chaotic dynamics of the Smale horseshoe type. 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.
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
5.
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
6.
Strategies for plane change of Earth orbits using lunar gravity and derived trajectories of family G
C. F. de Melo E. E. N. Macau O. C. Winter 《Celestial Mechanics and Dynamical Astronomy》2009,103(4):281-299
The dynamics of the circular restricted three-body Earth-Moon-particle problem predicts the existence of the retrograde periodic
orbits around the Lagrangian equilibrium point L1. Such orbits belong to the so-called family G (Broucke, Periodic orbits
in the restricted three-body problem with Earth-Moon masses, JPL Technical Report 32–1168, 1968) and starting from them it
is possible to define a set of trajectories that form round trip links between the Earth and the Moon. These links occur even
with more complex dynamical systems as the complete Sun-Earth-Moon-particle problem. One of the most remarkable properties
of these trajectories, observed for the four-body problem, is a meaningful inclination gain when they penetrate into the lunar
sphere of influence and accomplish a swing-by with the Moon. This way, when one of these trajectories returns to the proximities
of the Earth, it will be in a different orbital plane from its initial Earth orbit. In this work, we present studies that
show the possibility of using this property mainly to accomplish transfer maneuvers between two Earth orbits with different
altitudes and inclinations, with low cost, taking into account the dynamics of the four-body problem and of the swing-by as
well. The results show that it is possible to design a set of nominal transfer trajectories that require ΔV
Total less than conventional methods like Hohmann, bi-elliptic and bi-parabolic transfer with plane change. 相似文献
7.
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
8.
《New Astronomy》2019
The dynamics of the pseudo-Newtonian restricted four-body problem has been studied in the present paper, where the primaries have equal masses. The parametric variation of the existence as well as the position of the libration points are determined, when the value of the transition parameter ϵ ∈(0, 1]. The stability of these libration points has also been discussed. Our study reveals that the Jacobi constant as well as transition parameter ϵ have substantial effect on the regions of possible motion, where the fourth body is free to move. The multivariate version of Newton-Raphson iterative scheme is introduced for determining the basins of attraction in the configuration (x, y) plane. A systematic numerical investigation is executed to reveal the influence of the transition parameter on the topology of the basins of convergence. In parallel, the required number of iterations is also noted to show its correlations to the corresponding basins of convergence. It is unveiled that the evolution of the attracting regions in the pseudo-Newtonian restricted four-body problem is a highly complicated yet worth studying problem. 相似文献
9.
10.
Astronomy Letters - The elliptic case of restricted four-body problem with variable mass of infinitesimal body is studied here. The three primary bodies which are placed at the vertices of an... 相似文献
11.
Anoop Sivasankaran Bonnie A. Steves Winston L. Sweatman 《Celestial Mechanics and Dynamical Astronomy》2010,107(1-2):157-168
Several papers in the last decade have studied the Caledonian symmetric four-body problem (CSFBP), a restricted four-body system with a symmetrically reduced phase space. During these studies, difficulties have arisen when the system approaches a close encounter. These are due to collision singularities causing numerical integration algorithms to fail. In this paper, we give the full details of a regularisation approach that now enables us to study these close encounters and collision events. The resulting equations of motion can be efficiently integrated by a high-order integrator. The results from numerical testing of the algorithm verify that the regularisation is advantageous in preserving numerical stability. The effectiveness of the approach is illustrated for a range of CSFBP orbits. Numerical experiments show that the newly developed regularisation algorithm has excellent energy conservation properties. 相似文献
12.
In this paper we consider a restricted equilateral four-body problem where a particle of negligible mass is moving under the Newtonian gravitational attraction of three masses (called primaries) which move on circular orbits around their center of masses such that their configuration is always an equilateral triangle (Lagrangian configuration). We consider the case of two bodies of equal masses, which in adimensional units is the parameter of the problem. We study numerically the existence of families of unstable periodic orbits, whose invariant stable and unstable manifolds are responsible for the existence of homoclinic and heteroclinic connections, as well as of transit orbits traveling from and to different regions. We explore, for three different values of the mass parameter, what kind of transits and energy levels exist for which there are orbits with prescribed itineraries visiting the neighborhood of different primaries. 相似文献
13.
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
14.
An analytical study of the elliptic Sitnikov restricted four-body problem when all the primaries as source of same radiation pressure is presented. We find a solution, which is valid for small bounded oscillations in case of moderate eccentricity of the primary. We have linearized the equation of motion to obtain the Hill’s type equation. Using the Courant and Snyder transformation, Hill’s equation transformed into harmonic oscillator type equation. We have used the Lindstedt-Poincare perturbation method and again we have applied the Courant and Snyder transformation to obtain the final result. 相似文献
15.
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
16.
Joachim Schubart 《Celestial Mechanics and Dynamical Astronomy》2017,128(2-3):295-301
The paper refers to fictitious resonant orbits of planet type that surround both components of a binary system. In case of 16 studied examples a suitable choice of the starting values leads to a process of libration of special angular arguments and to an evolution with an at least temporary stay of the planet in the resonant orbit. The ratio of the periods of revolution of the binary and a planet is equal to 1:5. Eight orbits depend on the ratio 1:5 of the masses of the binary components, but two other ratios appear as well. The basis of this study is the planar, elliptic or circular restricted problem of three bodies, but remarks at the end of the text refer to a four-body problem. 相似文献
17.
Roberto Armellin Francesco Topputo 《Celestial Mechanics and Dynamical Astronomy》2006,96(3-4):289-309
A sixth-order accurate scheme is presented for the solution of ODE systems supplemented by two-point boundary conditions. The proposed integration scheme is a linear multi-point method of sixth-order accuracy successfully used in fluid dynamics and implemented for the first time in astrodynamics applications. A discretization molecule made up of just four grid points attains a O(h
6) accuracy which is beyond the first Dahlquist’s stability barrier. Astrodynamics applications concern the computation of libration point halo orbits, in the restricted three- and four-body models, and the design of an optimal control strategy for a low thrust libration point mission. 相似文献
18.
Luis Benet 《Celestial Mechanics and Dynamical Astronomy》2001,81(1-2):123-128
We consider the system of planetary rings with shepherds as a restricted three or four-body problem, neglecting interactions between ring particles. We show that the generic occurrence of rings for the case of rotating short-range potentials can be extended to the case of gravitational potentials. The consecutive collision periodic orbits created by saddle-center bifurcations are of central importance. 相似文献
19.
D. J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》1998,70(2):75-98
Starting from the four-body problem a generalization of both the restricted three-body problem and the Hill three-body problem
is derived. The model is time periodic and contains two parameters: the mass ratio ν of the restricted three-body problem
and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the restricted three-body problem
is recovered as m → 0 and the classical Hill three-body problem is recovered as ν → 0. This model also predicts motions described
by earlier researchers using specific models of the Earth–Moon–Sun system. An application of the current model to the motion
of a spacecraft in the Sun perturbed Earth–Moon system is made using Hill's Variation orbit for the motion of the Earth–Moon
system. The model is general enough to apply to the motion of an infinitesimal mass under the influence of any two primaries
which orbit a larger mass.
Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange
points are performed. Values of the parameter ν range in the neighborhood of the Earth–Moon value as the parameter m increases
from 0 to 0.195 at which point the Hill Variation orbit becomes unstable. Two families of planar periodic orbits are studied
in detail as the parameters m and ν vary. These families contain stable and unstable members in the plane and all have the
out-of-plane stability. The stable and unstable manifolds of the unstable periodic orbits are computed and found to be trapped
in a geometric area of phase space over long periods of time for ranges of the parameter values including the Earth–Moon–Sun
system.
This model is derived from the general four-body problem by rigorous application of the Hill and restricted approximations.
The validity of the Hill approximation is discussed in light of the actual geometry of the Earth–Moon–Sun system.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
20.
Minghu Tan Colin McInnes Matteo Ceriotti 《Celestial Mechanics and Dynamical Astronomy》2017,129(1-2):57-88
Near-Earth asteroids have attracted attention for both scientific and commercial mission applications. Due to the fact that the Earth–Moon \(\hbox {L}_{1}\) and \(\hbox {L}_{2}\) points are candidates for gateway stations for lunar exploration, and an ideal location for space science, capturing asteroids and inserting them into periodic orbits around these points is of significant interest for the future. In this paper, we define a new type of lunar asteroid capture, termed direct capture. In this capture strategy, the candidate asteroid leaves its heliocentric orbit after an initial impulse, with its dynamics modeled using the Sun–Earth–Moon restricted four-body problem until its insertion, with a second impulse, onto the \(\hbox {L}_{2}\) stable manifold in the Earth–Moon circular restricted three-body problem. A Lambert arc in the Sun-asteroid two-body problem is used as an initial guess and a differential corrector used to generate the transfer trajectory from the asteroid’s initial obit to the stable manifold associated with Earth–Moon \(\hbox {L}_{2}\) point. Results show that the direct asteroid capture strategy needs a shorter flight time compared to an indirect asteroid capture, which couples capture in the Sun–Earth circular restricted three-body problem and subsequent transfer to the Earth–Moon circular restricted three-body problem. Finally, the direct and indirect asteroid capture strategies are also applied to consider capture of asteroids at the triangular libration points in the Earth–Moon system. 相似文献