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1.
Marco Giancotti Mauro Pontani Paolo Teofilatto 《Celestial Mechanics and Dynamical Astronomy》2014,120(3):249-268
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
2.
Marco Giancotti Mauro Pontani Paolo Teofilatto 《Celestial Mechanics and Dynamical Astronomy》2012,114(1-2):55-76
Analysis and design of low-energy transfers to the Moon has been a subject of great interest for many decades. This paper is concerned with a topological study of such transfers, with emphasis to trajectories that allow performing lunar capture and those that exhibit homoclinic connections, in the context of the circular restricted three-body problem. A fundamental theorem stated by Conley locates capture trajectories in the phase space and can be condensed in a sentence: “if a crossing asymptotic orbit exists then near any such there is a capture orbit”. In this work this fundamental theoretical assertion is used together with an original cylindrical isomorphic mapping of the phase space associated with the third body dynamics. For a given energy level, the stable and unstable invariant manifolds of the periodic Lyapunov orbit around the collinear interior Lagrange point are computed and represented in cylindrical coordinates as tubes that emanate from the transformed periodic orbit. These tubes exhibit complex geometrical features. Their intersections correspond to homoclinic orbits and determine the topological separation of long-term lunar capture orbits from short-duration capture trajectories. The isomorphic mapping is proven to allow a deep insight on the chaotic motion that characterizes the dynamics of the circular restricted three-body, and suggests an interesting interpretation, and together corroboration, of Conley’s assertion on the topological location of lunar capture orbits. Moreover, an alternative three-dimensional representation of the phase space is profitably employed to identify convenient lunar periodic orbits that can be entered with modest propellant consumption, starting from the Lyapunov orbit. 相似文献
3.
Jeannette Heiligers Colin R. McInnes 《Celestial Mechanics and Dynamical Astronomy》2014,120(2):163-194
This paper introduces new families of Sun-centered non-Keplerian orbits (NKOs) that are constrained to a three-dimensional, cylindrical or spherical surface. As such, they are an extension to the well-known families of displaced NKOs that are confined to a two-dimensional plane. The cylindrical and spherical orbits are found by investigating the geometrically constrained spacecraft dynamics. By imposing further constraints on the orbit’s angular velocity and propulsive acceleration, the set of feasible orbits is defined. Additionally, the phase spaces of the orbits are explored and a numerical analysis is developed to find periodic orbits. The richness of the problem is further enhanced by considering both an inverse square acceleration law (mimicking solar electric propulsion) and a solar sail acceleration law to maintain the spacecraft on the three-dimensional surface. The wealth of orbits that these new families of NKOs generate allows for a range of novel space applications. 相似文献
4.
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
5.
The purpose of this paper is the semi-analytical computation of the bounded orbits of Hill’s equations, describing the relative
motion of two particles in their Keplerian motion around a central body. We have considered the case in which one of the particles
moves along a circular reference orbit. The solutions obtained are the generalisation of the usual periodic orbits obtained
for the linearised equations and also of the third-order solution computed by D.L. Richardson and J.W. Mitchell (2003). With
the algorithm presented, those orbits can be computed in a fast and efficient way up to a high-order in the in-plane and out-of-plane
amplitudes. 相似文献
6.
G. Voyatzis T. Kotoulas J. D. Hadjidemetriou 《Monthly notices of the Royal Astronomical Society》2009,395(4):2147-2156
The 2/1 resonant dynamics of a two-planet planar system is studied within the framework of the three-body problem by computing families of periodic orbits and their linear stability. The continuation of resonant periodic orbits from the restricted to the general problem is studied in a systematic way. Starting from the Keplerian unperturbed system, we obtain the resonant families of the circular restricted problem. Then, we find all the families of the resonant elliptic restricted three-body problem, which bifurcate from the circular model. All these families are continued to the general three-body problem, and in this way we can obtain a global picture of all the families of periodic orbits of a two-planet resonant system. The parametric continuation, within the framework of the general problem, takes place by varying the planetary mass ratio ρ. We obtain bifurcations which are caused either due to collisions of the families in the space of initial conditions or due to the vanishing of bifurcation points. Our study refers to the whole range of planetary mass ratio values [ρ∈ (0, ∞)] and, therefore we include the passage from external to internal resonances. Thus, we can obtain all possible stable configurations in a systematic way. As an application, we consider the dynamics of four known planetary systems at the 2/1 resonance and we examine if they are associated with a stable periodic orbit. 相似文献
7.
F. J. T. Salazar C. F. de Melo E. E. N. Macau O. C. Winter 《Celestial Mechanics and Dynamical Astronomy》2012,114(1-2):201-213
An alternative transfer strategy to send spacecraft to stable orbits around the Lagrangian equilibrium points L4 and L5 based in trajectories derived from the periodic orbits around L1 is presented in this work. The trajectories derived, called Trajectories G, are described and studied in terms of the initial generation requirements and their energy variations relative to the Earth through the passage by the lunar sphere of influence. Missions for insertion of spacecraft in elliptic orbits around L4 and L5 are analysed considering the restricted three-body problem Earth–Moon-particle and the results are discussed starting from the thrust, time of flight and energy variation relative to the Earth. 相似文献
8.
John D. Hadjidemetriou Th. Christides 《Celestial Mechanics and Dynamical Astronomy》1975,12(2):175-187
A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm 3 of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom 1:m 2:m 3?5:5:3. From that point on the family continues for decreasing massesm 3 until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem. 相似文献
9.
We continue to analyze the periodic solutions of the singly averaged Hill problem. We have numerically constructed the families of solutions that correspond to periodically evolving satellite orbits for arbitrary initial values of their eccentricities and inclinations to the plane of motion of the perturbing body. The solutions obtained are compared with the numerical solutions of the rigorous (nonaveraged) equations of the restricted circular three-body problem. In particular, we have constructed a periodically evolving orbit for which the well-known Lidov-Kozai mechanism manifests itself, just as in the doubly averaged problem. 相似文献
10.
《New Astronomy》2022
The famous three-body problem can be traced back to Newton in 1687, but quite few families of periodic orbits were found in 300 years thereafter. In this paper, we propose an effective approach and roadmap to numerically gain planar periodic orbits of three-body systems with arbitrary masses by means of machine learning based on an artificial neural network (ANN) model. Given any a known periodic orbit as a starting point, this approach can provide more and more periodic orbits (of the same family name) with variable masses, while the mass domain having periodic orbits becomes larger and larger, and the ANN model becomes wiser and wiser. Finally we have an ANN model trained by means of all obtained periodic orbits of the same family, which provides a convenient way to give accurate enough predictions of periodic orbits with arbitrary masses for physicists and astronomers. It suggests that the high-performance computer and artificial intelligence (including machine learning) should be the key to gain periodic orbits of the famous three-body problem. 相似文献
11.
P. Delibaltas 《Celestial Mechanics and Dynamical Astronomy》1983,29(2):191-204
In the general three-body problem, in a rotating frame of reference, a symmetric periodic solution with a binary collision is determined by the abscissa of one body and the energy of the system. For different values of the masses of the three bodies, the symmetric periodic collision orbits form a two-parametric family. In the case of equal masses of the two bodies and small mass of the third body, we found several symmetric periodic collision orbits similar to the corresponding orbits in the restricted three-body problem. Starting with one symmetric periodic collision orbit we obtained two families of such orbits. Also starting with one collision orbit in the Sun-Jupiter-Saturn system we obtained, for a constant value of the mass ratio of two bodies, a family of symmetric periodic collision orbits. 相似文献
12.
We study the orbits and manifolds near the equilibrium points of a rotating asteroid. The linearised equations of motion relative to the equilibrium points in the gravitational field of a rotating asteroid, the characteristic equation and the stable conditions of the equilibrium points are derived and discussed. First, a new metric is presented to link the orbit and the geodesic of the smooth manifold. Then, using the eigenvalues of the characteristic equation, the equilibrium points are classified into 8 cases. A theorem is presented and proved to describe the structure of the submanifold as well as the stable and unstable behaviours of a massless test particle near the equilibrium points. The linearly stable, the non-resonant unstable, and the resonant equilibrium points are discussed. There are three families of periodic orbits and four families of quasi-periodic orbits near the linearly stable equilibrium point. For the non-resonant unstable equilibrium points, there are four relevant cases; for the periodic orbit and the quasi-periodic orbit, the structures of the submanifold and the subspace near the equilibrium points are studied for each case. For the resonant equilibrium points, the dimension of the resonant manifold is greater than 4, and we find at least one family of periodic orbits near the resonant equilibrium points. As an application of the theory developed here, we study relevant orbits for the asteroids 216 Kleopatra, 1620 Geographos, 4769 Castalia and 6489 Golevka. 相似文献
13.
R. C. Calleja E. J. Doedel A. R. Humphries A. Lemus-Rodríguez E. B. Oldeman 《Celestial Mechanics and Dynamical Astronomy》2012,114(1-2):77-106
We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the circular restricted three-body problem (CR3BP), which models the motion of a satellite in an Earth–Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar vertical and halo orbits. We compute the unstable manifolds of selected vertical and halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits are of potential interest in space mission design. 相似文献
14.
V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》1980,21(3):291-309
The three-dimensional general three-body problem is formulated suitably for the numerical determination of periodic orbits either directly or by continuation from the three-dimensional periodic orbits of the restricted problem. The symmetry properties of the equations of motion are established and the algorithms for the numerical determination of families of periodic orbits are outlined. A normalization scheme based on the concept of the invariable plane is introduced to simplify the process. All three types of symmetric orbit, as well as the general type of asymmetric orrbit, are considered. Many threedimmensional p periodic orbits are given. 相似文献
15.
We analyze nearly periodic solutions in the plane problem of three equal-mass bodies by numerically simulating the dynamics of triple systems. We identify families of orbits in which all three points are on one straight line (syzygy) at the initial time. In this case, at fixed total energy of a triple system, the set of initial conditions is a bounded region in four-dimensional parameter space. We scan this region and identify sets of trajectories in which the coordinates and velocities of all bodies are close to their initial values at certain times (which are approximately multiples of the period). We classify the nearly periodic orbits by the structure of trajectory loops over one period. We have found the families of orbits generated by von Schubart’s stable periodic orbit revealed in the rectilinear three-body problem. We have also found families of hierarchical, nearly periodic trajectories with prograde and retrograde motions. In the orbits with prograde motions, the trajectory loops of two close bodies form looplike structures. The trajectories with retrograde motions are characterized by leafed structures. Orbits with central and axial symmetries are identified among the families found. 相似文献
16.
C. Zagouras 《Astrophysics and Space Science》1977,50(2):383-407
A number of partly known families of symmetric three-dimensional periodic orbits of the restricted three-body (=0.4) problem are numerically continued in both ends until they terminate with orbits in the plane of motion of the primaries. The families of plane symmetric periodic orbits from which they bifurcate are identified and many orbit illustrations are given. 相似文献
17.
18.
We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches.
We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides
the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative
periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with
binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant
families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As
the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between
triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge
to “previously unknown” periodic orbits in the planar problem. 相似文献
19.
A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when
a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic
solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of
these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of
all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist
very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections,
that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion. 相似文献
20.
Winston L. Sweatman 《Celestial Mechanics and Dynamical Astronomy》2006,94(1):37-65
We locate members of an important category of periodic orbits in the Newtonian four-body problem. These systems perform an interplay motion similar to that of the periodic three-body orbit discovered by Schubart. Such orbits, when stable, have been shown to be a key feature and influence on the dynamics of few-body systems. We consider the restricted case where the masses are collinear and are distributed symmetrically about their centre of mass. A family of orbits is generated from the known (three-dimensionally) unstable equal masses case by varying the mass ratio, whilst maintaining the symmetry. The stability of these orbits to perturbation is studied using linear stability analysis, analytical approximation of limiting cases and nonlinear simulation. We answer the natural question: are there any stable periodic orbits of this kind? Three ranges of the mass ratio are found to have stable orbits and three ranges have unstable orbits for three-dimensional motion. The systems closely resemble their three-body counterparts. Here the family of interplay orbits is simpler requiring just one parameter to characterise the mass ratio. Our results provide a further insight into three-body orbits studied previously. 相似文献