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1.
Edward Belbruno Jaume Llibre Mercè Ollé 《Celestial Mechanics and Dynamical Astronomy》1994,60(1):99-129
In this paper we deal with the circular Sitnikov problem as a subsystem of the three-dimensional circular restricted three-body problem. It has a first analytical part where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; and we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. We compare our results with the previous ones of other authors on this problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to 1/2 are also described. 相似文献
2.
The Sitnikov configuration is a special case of the restricted three-body problem where the two primaries are of equal masses
and the third body of a negligible mass moves along a straight line perpendicular to the orbital plane of the primaries and
passes through their center of mass. It may serve as a toy model in dynamical astronomy, and can be used to study the three-dimensional
orbits in more applicable cases of the classical three-body problem. The present paper concerns the straight-line oscillations
of the Sitnikov family of the photogravitational circular restricted three-body problem as well as the associated families
of three-dimensional periodic orbits. From the stability analysis of the Sitnikov family and by using appropriate correctors
we have computed accurately 49 critical orbits at which families of 3D periodic orbits of the same period bifurcate. All these
families have been computed in both cases of equal and non-equal primaries, and consist entirely of unstable orbits. They
all terminate with coplanar periodic orbits. We have also found 35 critical orbits at which period doubling bifurcations occur.
Several families of 3D periodic orbits bifurcating at these critical Sitnikov orbits have also been given. These families
contain stable parts and close upon themselves containing no coplanar orbits. 相似文献
3.
4.
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
5.
We consider the photogravitational restricted three-body problem with oblateness and study the Sitnikov motions. The family of straight line oscillations exists only in the case where the primaries are of equal masses as in the classical Sitnikov problem and have the same oblateness coefficients and radiation factors. A perturbation method based on Floquet theory is applied in order to study the stability of the motion and critical orbits are determined numerically at which families of three-dimensional periodic orbits of the same or double period bifurcate. Many of these families are computed. 相似文献
6.
Zsolt Sàndor Róbert Balla Ferenc Téger Bálint Érdi 《Celestial Mechanics and Dynamical Astronomy》2001,79(1):29-40
We applied the method of the short time Lyapunov indicators to the planar circular and to the planar elliptic restricted three-body problem in order to study the structure of the phase space in some selected regions. In the circular case we computed the short-time averages of the stretching numbers to distinguish between regular and chaotic domains of the phase space. The results obtained in this way are in good agreement with the corresponding Poincaré's surface of sections. Using the short time Lyapunov indicators we determined the detailed structure of the phase space in the semi-major axis-eccentricity plane of the test particle both in the circular and in the elliptic restricted problem (in the latter case for some values of the eccentricities of the primaries) and we studied the main features of the phase space.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
7.
We perform an analysis of the dynamics of the circular, restricted, planar three-body problem under the effect of different
kinds of dissipation (linear, Stokes and Poynting–Robertson drags). Since the problem is singular, we implement a regularization
technique in the style of Levi–Civita. The effect of the dissipation is often to decrease the semi-major axis; as a consequence
the minor body collides with one of the primaries. In general, it is quite difficult to find non-collision orbits using random
initial conditions. However, by means of the computation of the Fast Lyapunov Indicators (FLI), we obtain a global view of
the dynamics. Precisely, we detect the regions of the phase space potentially belonging to basins of attraction. This investigation
provides information on the different regions of the phase space, showing both collision and non-collision trajectories. Moreover,
we find periodic orbit attractors for the case of linear and Stokes drags, while in the case of the Poynting–Robertson effect
no other attractors are found beside the primaries, unless a fourth body is added to counterbalance the dissipative effect. 相似文献
8.
Marc Fouchard Elena Lega Christiane Froeschlé Claude Froeschlé 《Celestial Mechanics and Dynamical Astronomy》2002,83(1-4):205-222
It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings. 相似文献
9.
In this paper, we deal with a Hill’s equation, depending on two parameters \(e\in [0,1)\) and \(\varLambda >0\), that has applications to some problems in Celestial Mechanics of the Sitnikov type. Due to the nonlinearity of the eccentricity parameter e and the coexistence problem, the stability diagram in the \((e,\varLambda )\)-plane presents unusual resonance tongues emerging from points \((0,(n/2)^2),\ n=1,2,\ldots \) The tongues bounded by curves of eigenvalues corresponding to \(2\pi \)-periodic solutions collapse into a single curve of coexistence (for which there exist two independent \(2\pi \)-periodic eigenfunctions), whereas the remaining tongues have no pockets and are very thin. Unlike most of the literature related to resonance tongues and Sitnikov-type problems, the study of the tongues is made from a global point of view in the whole range of \(e\in [0,1)\). Indeed, an interesting behavior of the tongues is found: almost all of them concentrate in a small \(\varLambda \)-interval [1, 9 / 8] as \(e\rightarrow 1^-\). We apply the stability diagram of our equation to determine the regions for which the equilibrium of a Sitnikov \((N+1)\)-body problem is stable in the sense of Lyapunov and the regions having symmetric periodic solutions with a given number of zeros. We also study the Lyapunov stability of the equilibrium in the center of mass of a curved Sitnikov problem. 相似文献
10.
Using the famous Sundman inequality, we have constructed for the first time the surfaces for the general three-body problem that we suggest calling Sundman surfaces. These surfaces are a generalization of the widely known Hill surfaces in the restricted circular three-body problem. The Sundman surfaces are constructed in a rectangular coordinate system that uses the mutual distances between the bodies as the Cartesian rectangular coordinates. The singular points of the family of these surfaces have been determined. The possible and impossible regions of motion of the bodies have been constructed in the space of mutual distances. We have shown the existence of Hill stable motions and established sufficient criteria for Hill stability of motions. Some of the astronomical applications are considered. 相似文献
11.
The spatial restricted rhomboidal five-body problem, or shortly, SRRFBP, is a five body problem in which four positive masses, called the primaries, move two by two in coplanar circular motions with the center of mass fixed at the origin such that their configuration is always a rhombus, the fifth mass being negligible and not influencing the motion of the four primaries. The Hamiltonian function that governs the motion of the fifth mass is derived and has three degrees of freedom depending periodically on time. Using a synodical system of coordinates, we fix the primaries in order to eliminate the time dependence. With the help of the Hamiltonian structure, we characterize the regions of possible motion. The vertical $z$ axis is invariant and we study what we call the rhomboidal Sitnikov problem. Unlike the classical Sitnikov problem, no chaos exists and the behavior of the fifth mass is quite predictable, periodic solutions of arbitrary long periods are shown to exist and we study numerically their linear horizontal stability. 相似文献
12.
Alessandro Morbidelli Claude Froeschlé 《Celestial Mechanics and Dynamical Astronomy》1995,63(2):227-239
On the basis of the general theory of Hamiltonian systems, we consider the relationship between Lyapunov times and macroscopic diffusion times. We find out that there are two regimes: the Nekhoroshev regime and the resonant overlapping regime. In the first case the diffusion time is exponentially long with respect to Lyapunov times. In the second case, the relationship is polynomial although we do not find any theoretical reason for the existence of a universal power law. We show numerical evidences which confirm our theoretical considerations. 相似文献
13.
Applications of lagrangian coherent structures to expression of invariant manifolds in astrodynamics
This paper investigates the relationship between invariant manifold and Lagrangian coherent structure (LCS) in dynamical systems. LCS is defined as the ridge of finite-time Lyapunov exponent (FTLE) field, and is proving to be excellent platform for studies of stable and unstable manifold in flows with arbitrary time dependence. In this study, the LCS tool is applied to autonomous systems, simple pendulum and planar circular restricted three-body problem (PCR3BP), and also non-autonomous ones, double-gyre flow and bicircular problem (BCP). A comparison between LCS and invariant manifold is presented. 相似文献
14.
B. F. Villac 《Celestial Mechanics and Dynamical Astronomy》2008,102(1-3):29-48
Fast Lyapunov Indicator (FLI) maps are presented as a tool for solving spacecraft preliminary trajectory design problems in multi-body environments with long-term stability requirements. In particular, the FLI maps are shown to provide a global overview of the dynamics in the restricted three-body problem that can guide mission designers in selecting long-term stable regions of phase space which are inherently more robust to model parameter perturbations. The FLI is also shown to numerically detect the normally hyperbolic manifolds associated with unstable periodic orbits. These, in turn, provide a global map of the principal heteroclinic connections between the various resonance regions which form the basic backbone of dynamical transfers design. Examples of maps and transfers are provided in the restricted three-body problem modeling the Jupiter–Europa system. 相似文献
15.
对一类具非零角动量的平面三体系统研究其三体构形对系统演化的影响．根据Agekian和Anosova提出的构形图(homology map)，三体系统按其构形特点分属于4个不同的区域．通过数值计算，考察了初始位置位于不同区域中的构形颗粒(homolgydrop)的演化，并就有关性质与Heinamaki等人研究的角动量为零的三体系统作了比较指出，构形颗粒的组成系统全部发生解体的时间在L区域最早，H区域最晚，这与零角动量系统不同．还对4个区域的三体系统的寿命进行了统计分析，得到了各区域中末解体的系统数随时间指数衰减的函数关系． 相似文献
16.
《New Astronomy》2021
In the present work, we shall show an exhaustive numerical analysis of the dynamical behavior of the test particle in the modified restricted three-body problem with variable mass in which the extra effect of the three-body interaction is also taken into account. This additional force ingredient appears in the potential of the classical problem as a new extra term. As the main results, we determine the motion for test particle, under the effect of three-body interaction, which varies its mass according to Jeans’ law (Jeans (1928)). The number and existence of the libration points along with their stability have been investigated as the function of value of the parameters which occur due to variable mass of the test particle. Moreover, the regions of the possible motion are also unveiled where the test particle is free to move. Furthermore, the multivariate version of the Newton-Raphson (NR) iterative scheme is used to determine the outcomes of the used parameters on the topology of the basins of convergence (BoC) linked to the libration points. The numerical analysis shows that the topology of the basins of convergence linked with the libration points is highly influenced by the used parameters. Moreover, we perform a systematic analysis to unveil how the regions of convergence are related with the number of required iterations and also with the corresponding probability distributions. 相似文献
17.
We deal with the problem of a zero mass body oscillating perpendicular to a plane in which two heavy bodies of equal mass orbit each other on Keplerian ellipses. The zero mass body intersects the primaries plane at the systems barycenter. This problem is commonly known as theSitnikov Problem. In this work we are looking for a first integral related to the oscillatory motion of the zero mass body. This is done by first expressing the equation of motion by a second order polynomial differential equation using a Chebyshev approximation techniques. Next we search for an autonomous mapping of the canonical variables over one period of the primaries. For that we discretize the time dependent coefficient functions in a certain number of Dirac Delta Functions and we concatenate the elementary mappings related to the single Delta Function Pulses. Finally for the so obtained polynomial mapping we look for an integral also in polynomial form. The invariant curves in the two dimensional phase space of the canonical variables are investigated as function of the primaries eccentricity and their initial phase. In addition we present a detailed analysis of the linearized Sitnikov Problem which is valid for infinitesimally small oscillation amplitudes of the zero mass body. All computations are performed automatically by the FORTRAN program SALOME which has been designed for stability considerations in high energy particle accelerators. 相似文献
18.
The stability of co-orbital motions is investigated in such exoplanetary systems, where the only known giant planet either moves fully in the habitable zone, or leaves it for some part of its orbit. If the regions around the triangular Lagrangian points are stable, they are possible places for smaller Trojan-like planets. We have determined the nonlinear stability regions around the Lagrangian point L4 of nine exoplanetary systems in the model of the elliptic restricted three-body problem by using the method of the relative Lyapunov indicators. According to our results, all systems could possess small Trojan-like planets. Several features of the stability regions are also discussed. Finally, the size of the stability region around L4 in the elliptic restricted three-body problem is determined as a function of the mass parameter and eccentricity. 相似文献
19.
20.
In this paper, we show the important role of chaotic transients in Celestial Mechanics through the Sitnikov problem. We compare
the two kinds of chaos, permanent and transient, and provide the chaotic saddle of the Sitnikov problem giving also some important
quantitative properties of this fractal set. Additionally, we present a link between the stickiness effect of tori and chaotic
scattering. 相似文献