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1.
In this paper the effect of solar radiation pressure on the location and stability of the five Lagrangian points is studied, within the frame of elliptic restricted three-body problem, where the primaries are the Sun and Jupiter acting on a particle of negligible mass. We found that the radiation pressure plays the rule of slightly reducing the effective mass of the Sun and changes the location of the Lagrangian points. New formulas for the location of the collinear libration points were derived. For large values of the force ratio β, we found that at β=0.12, the collinear point L3 is stable and some families of periodic orbits can be drawn around it.  相似文献   

2.
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   

3.
Starting from 80 families of low-energy fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), we obtain by analytical continuation 11 periodic orbits and 25 periodic arcs with similar properties in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). A novel and very simple procedure is introduced giving the solar phases at which to attempt continuation. Detailed numerical results for each periodic orbit and arc found are given, including their stability parameters and minimal distances to the Earth and Moon. The periods of these orbits are between 2.5 and 5 synodic months, their energies are among the lowest possible to achieve an Earth–Moon transfer, and they show a diversity of circumlunar trajectories, making them good candidates for missions requiring repeated passages around the Earth and the Moon with close approaches to the last.  相似文献   

4.
Starting from the identification and classification of a family of fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), and using analytic continuation techniques, we find two unstable periodic orbits in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). The orbits found perform periodic Earth–Moon transfers with a period of approximately 29.5 days.  相似文献   

5.
Consider the Earth-Moon-particle system as a Restricted Three Body Problem. There are two equilateral libration points. In the actual world system, those points are no longer relative equilibrium points mainly due to the effect of the Sun and to the noncircular motion of the Moon around the Earth. In this paper we present the problem as a perturbation of the RTBP and we look for the dynamical equivalent of L 4,5. It turns out to be a quasiperiodic orbit. It is obtained for a simplified model but the procedure to obtain it is general and can be carried out with an additional computational effort.  相似文献   

6.
The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L4 in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μR (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic orbits and the 3D nearby tori are also described.  相似文献   

7.
We study numerically the restricted five-body problem when some or all the primary bodies are sources of radiation. 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 are given. We found that the number of the collinear equilibrium points of the problem depends on the mass parameter β and the radiation factors q i , i=0,…,3. The stability of the equilibrium points are also studied. Critical masses associated with the number of the equilibrium points and their stability are given. The network of the families of simple symmetric periodic orbits, vertical critical periodic solutions and the corresponding bifurcation three-dimensional families when the mass parameter β and the radiation factors q i vary are illustrated. Series, with respect to the mass (and to the radiation) parameter, of critical periodic orbits are calculated.  相似文献   

8.
WSO／UV(世界空间紫外天文台)以及监测太阳活动的特殊探测器(Solar Sentinel)都需要在日-地(月)系的平动点附近运行，且相对日-地(月)系要求其几何位置几乎保持不变，因此有必要阐明平动点的动力学特征及其附近的运动状况。基于这一点，对限制性三体模型下，日-地(月)系中平动点附近扰动运动的稳定性作了详尽的分析，尤其讨论了共线平动点具有不稳定动力学特征时，如何使WSO／UV这类空间探测器保持在其附近的情况；同时阐明了轨道保持不变的条件和相应的轨控措施。  相似文献   

9.
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   

10.
This paper deals with the Restricted Three Body Problem (RTBP) in which we assume that the primaries are radiation sources and the influence of the radiation pressure on the gravitational forces is considered; in particular, we are interested in finding families of periodic orbits under theses forces. By means of some modifications to the method of numerical continuation of natural families of periodic orbits, we find several families of periodic orbits, both in two and three dimensions. As starters for our method we use some known periodic orbits in the classical RTBP. This revised version was published online in August 2006 with corrections to the Cover Date.  相似文献   

11.
We study the equilibrium points and the zero-velocity curves of Chermnykh’s problem when the angular velocity ω varies continuously and the value of the mass parameter is fixed. The planar symmetric simple-periodic orbits are determined numerically and they are presented for three values of the parameter ω. The stability of the periodic orbits of all the families is computed. Particularly, we explore the network of the families when the angular velocity has the critical value ω = 2√2 at which the triangular equilibria disappear by coalescing with the collinear equilibrium point L1. The analytic determination of the initial conditions of the family which emanate from the Lagrangian libration point L1 in this case, is given. Non-periodic orbits, as points on a surface of section, providing an outlook of the stability regions, chaotic and escape motions as well as multiple-periodic orbits, are also computed. Non-linear stability zones of the triangular Lagrangian points are computed numerically for the Earth–Moon and Sun–Jupiter mass distribution when the angular velocity varies.  相似文献   

12.
Due to various perturbations, the collinear libration points of the real Earth–Moon system are not equilibrium points anymore. Under the assumption that the Moon’s motion is quasi-periodic, special quasi-periodic orbits called dynamical substitutes exist. These dynamical substitutes replace the geometrical collinear libration points as time-varying equilibrium points. In the paper, the dynamical substitutes of the three collinear libration points in the real Earth–Moon system are computed. For the points L 1 and L 2, linearized motions around the dynamical substitutes are described, and the variational equations of the dynamical substitutes are reduced to a form with a near constant coefficient matrix. Then higher order analytical formulae of the central manifolds are constructed. Using these analytical solutions as initial seeds, Lissajous orbits and halo orbits are computed with numerical algorithms.  相似文献   

13.
This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits.  相似文献   

14.
The present research was motivated by the recent discovery of planets around binary stars. Our initial intention was thus to investigate the 3-dimensional nearly circular periodic orbits of the circular restricted problem of three bodies; more precisely Stromgren's class L, (direct) and class m, (retrograde). We started by extending several of Hénon's vertical critical orbits of these 2 classes to three dimensions, looking especially for orbits which are near circular and have stable characteristic exponents.We discovered early on that the periodic orbits with the above two qualifications are fairly rare and we decided thus to undertake a systematic exploration, limiting ourselves to symmetric periodic orbits. However, we examined all 16 possible symmetry cases, trying 10000 sets of initial values for periodicity in each case, thus 160000 integrations, all with z o or o equal to 0.1 This gave us a preliminary collection of 171 periodic orbits, all fairly near the xy-plane, thus with rather low inclinations. Next, we integrated a second similar set of 160000 cases with z o or o equal to 0.5, in order to get a better representation of the large inclinations. This time, we found 167 periodic orbits, but it was later discovered that at least 152 of them belong to the same families as the first set with 0.1Our paper quickly describes the definition of the problem, with special emphasis on the symmetry properties, especially for the case of masses with equal primaries. We also allow a section to describe our approach to stability and characteristic exponents, following our paper on this subject, (Broucke, 1969). Then we describe our numerical results, as much as space permits in the present paper.We found basically only about a dozen families with sizeable segments of simple stable periodic orbits. Some of them are around one of the two stars only but we do not describe them here because of a lack of space. We extended about 170 periodic orbits to families of up to 500 members, (by steps of 0.005 in the parameter), although, in many cases, we do not know the real end of the families. We also give an overview of the different types of periodic orbits that are most often encountered. We describe some of the rather strange orbits, (some of which are actually stable).  相似文献   

15.
The present study deals with numerical modeling of the elliptic restricted three-body problem as well as of the perturbed elliptic restricted three-body (Earth-Moon-Satellite) problem by a fourth body (Sun). Two numerical algorithms are established and investigated. The first is based on the method of the series solution of the differential equations and the second is based on a 5th-order Runge-Kutta method. The applications concern the solution of the equations and integrals of motion of the circular and elliptical restricted three-body problem as well as the search for periodic orbits of the natural satellites of the Moon in the Earth-Moon system in both cases in which the Moon describes circular or elliptical orbit around the Earth before the perturbations induced by the Sun. After the introduction of the perturbations in the Earth-Moon-Satellite system the motions of the Moon and the Satellite are studied with the same initial conditions which give periodic orbits for the unperturbed elliptic problem.  相似文献   

16.
A method is described for the numerical determination of families of periodic orbits in the planar restricted problem of three bodies. The families are sought in their representation as curves in a two-dimensional space of parameters. A grid search is applied to the study of the evolution of satellite motion when the mass parameter is varied. Only that part of the space of parameters is investigated for which one of them, the relative energy constant, takes values larger than that corresponding to the inner Lagrangian pointL 2. Critical values of the mass parameter are determined for which new families of simple or double periodic orbits appear inside the closed ovals of zero velocity.  相似文献   

17.
In the present study an investigation of the collision orbits of natural satellites of the Moon (considered to be of finite dimensions) is developed, and the tendency of natural satellites of the Moon to collide on the visible or the far side of the Moon is studied. The collision course of the satellite is studied up to its impact on the lunar surface for perturbations of its initial orbit arbitrarily induced, for example, by the explosion of a meteorite. Several initial conditions regarding the position of the satellite to collide with the Moon on its near (visible) or far (invisible) side is examined in connection to the initial conditions and the direction of the motion of the satellite. The distribution of the lunar craters-originating impact of lunar satellites or celestial bodies which followed a course around the Moon and lost their stability - is examined. First, we consider the planar motion of the natural satellite and its collision on the Moon's surface without the presence of the Earth and Sun. The initial velocities of the satellite are determined in such a way so its impact on the lunar surface takes place on the visible side of the Moon. Then, we continue imparting these velocities to the satellite, but now in the presence of the Earth and Sun; and study the forementioned impacts of the satellites but now in the Earth-Moon-Satellite system influenced also by the Sun. The initial distances of the satellite are taken as the distances which have been used to compute periodic orbits in the planar restricted three-body problem (cf. Gousidou-Koutita, 1980) and its direction takes different angles with the x-axis (Earth-Moon axis). Finally, we summarise the tendency of the satellite's impact on the visible or invisible side of the Moon.  相似文献   

18.
We study numerically the asymmetric periodic orbits which emanate from the triangular equilibrium points of the restricted three-body problem under the assumption that the angular velocity ω varies and for the Sun–Jupiter mass distribution. The symmetric periodic orbits emanating from the collinear Lagrangian point L 3, which are related to them, are also examined. The analytic determination of the initial conditions of the long- and short-period Trojan families around the equilibrium points, is given. The corresponding families were examined, for a combination of the mass ratio and the angular velocity (case of equal eigenfrequencies), and also for the critical value ω = 2
, at which the triangular equilibria disappear by coalescing with the inner collinear equilibrium point L 1. We also compute the horizontal and the vertical stability of these families for the angular velocity parameter ω under consideration. Series of horizontal–critical periodic orbits of the short-Trojan families with the angular velocity ω and the mass ratio μ as parameters, are given.  相似文献   

19.
This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L 1, L 2 and L 3 correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations.  相似文献   

20.
Several families of periodic orbits exist in the context of the circular restricted three-body problem. This work studies orbital motion of a spacecraft among these periodic orbits in the Earth–Moon system, using the planar circular restricted three-body problem model. A new cylindrical representation of the spacecraft phase space (i.e., position and velocity) is described, and allows representing periodic orbits and the related invariant manifolds. In the proximity of the libration points, the manifolds form a four-fold surface, if the cylindrical coordinates are employed. Orbits departing from the Earth and transiting toward the Moon correspond to the trajectories located inside this four-fold surface. The isomorphic mapping under consideration is also useful for describing the topology of the invariant manifolds, which exhibit a complex geometrical stretch-and-folding behavior as the associated trajectories reach increasing distances from the libration orbit. Moreover, the cylindrical representation reveals extremely useful for detecting periodic orbits around the primaries and the libration points, as well as the possible existence of heteroclinic connections. These are asymptotic trajectories that are ideally traveled at zero-propellant cost. This circumstance implies the possibility of performing concretely a variety of complex Earth–Moon missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining a suitable, convenient end-of-life strategy for spacecraft placed in any of the unstable orbits. The final disposal orbit is an externally confined trajectory, never approaching the Earth or the Moon, and can be entered by means of a single velocity impulse (of modest magnitude) along the right unstable manifold that emanates from the Lyapunov orbit at \(L_2\) .  相似文献   

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