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1.
Priscilla Sousa-Silva Maisa O. Terra Matteo Ceriotti 《Astrophysics and Space Science》2018,363(10):210
This contribution deals with fast Earth–Moon transfers with ballistic capture in the patched three-body model. We compute ensembles of preliminary solutions using a model that takes into account the relative inclination of the orbital planes of the primaries. The ballistic capture orbits around the Moon are obtained relying on the hyperbolic invariant structures associated to the collinear Lagrangian points of the Earth–Moon system, and the Sun–Earth system portion of the transfers are quasi-periodic orbits obtained by a genetic algorithm. The trajectories are designed to be good initial guesses to search optimal cost-efficient short-time Earth–Moon transfers with ballistic capture in more realistic models. 相似文献
2.
In this paper, distant quasi-periodic orbits around Mercury are studied for future Mercury missions. All of these orbits have relatively large sizes, with their altitudes near or above the Mercury sphere of influence. The research is carried out in the framework of the elliptic restricted three-body problem (ER3BP) to account for the planet’s non-negligible orbital eccentricity. Retrograde and prograde quasi-periodic trajectories in the planar ER3BP are generalized from periodic orbits in the CR3BP by the homotopy algorithm, and the shape evolution of such quasi-periodic trajectories around Mercury is investigated. Numerical simulations are performed to evaluate the stability of these distant orbits in the long term. These two classes of orbits present different characteristics: retrograde orbits can maintain shape stability with a large size, although the trajectories in some regions may oscillate with larger amplitudes; for prograde orbits, the range of existence is much smaller, and their trajectories easily move away from the vicinity of Mercury when the orbits become larger. Distant orbits can be used to explore the space environment in the vicinity of Mercury, and some orbits can be taken as transfer orbits for low-cost Mercury return missions or other programs for their high maneuverability. 相似文献
3.
P. C. Kammeyer 《Celestial Mechanics and Dynamical Astronomy》1976,14(2):159-165
This paper reports the existence of three types of satellite orbit periodic in coordinates rotating with the Earth. Results on linear orbital stability are presented. Also included is a survey of the author's results on quasi-periodic orbits. 相似文献
4.
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
5.
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
6.
K. Uldall Kristiansen P. L. Palmer M. Roberts 《Celestial Mechanics and Dynamical Astronomy》2010,106(4):371-390
This paper builds upon the work of Palmer and Imre exploring the relative motion of satellites on neighbouring Keplerian orbits.
We make use of a general geometrical setting from Hamiltonian systems theory to obtain analytical solutions of the variational
Kepler equations in an Earth centred inertial coordinate frame in terms of the relevant conserved quantities: relative energy,
relative angular momentum and the relative eccentricity vector. The paper extends the work on relative satellite motion by
providing solutions about any elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum
case. The geometrical framework assists the design of complex formation flying trajectories. This is demonstrated by the construction
of a tetrahedral formation, described through the relevant conserved quantities, for which the satellites are on highly eccentric
orbits around the Sun to visit the Kuiper belt. 相似文献
7.
Using alternative independent variables in lieu of time has important advantages when propagating the partial derivatives of the trajectory. This paper focuses on spacecraft relative motion, but the concepts presented here can be extended to any problem involving the variational equations of orbital motion. A usual approach for modeling the relative dynamics is to evaluate how the reference orbit changes when modifying the initial conditions slightly. But when the time is a mere dependent variable, changes in the initial conditions will result in changes in time as well: a time delay between the reference and the neighbor solution will appear. The theory of asynchronous relative motion shows how the time delay can be corrected to recover the physical sense of the solution and, more importantly, how this correction can be used to improve significantly the accuracy of the linear solutions to relative motion found in the literature. As an example, an improved version of the Clohessy-Wiltshire (CW) solution is presented explicitly. The correcting terms are extremely compact, and the solution proves more accurate than the second and even third order CW equations for long propagations. The application to the elliptic case is also discussed. The theory is not restricted to Keplerian orbits, as it holds under any perturbation. To prove this statement, two examples of realistic trajectories are presented: a pair of spacecraft orbiting the Earth and perturbed by a realistic force model; and two probes describing a quasi-periodic orbit in the Jupiter-Europa system subject to third-body perturbations. The numerical examples show that the new theory yields reductions in the propagation error of several orders of magnitude, both in position and velocity, when compared to the linear approach. 相似文献
8.
Ming Xu Jiamin Zhu Tian Tan Shijie Xu 《Celestial Mechanics and Dynamical Astronomy》2012,113(4):403-433
The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J 2 perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J 2-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories. 相似文献
9.
We explore the effect of oblateness of Saturn (more massive primary) on the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. First order interior and exterior mean motion resonances are located. The effect of oblateness is studied on the location, nature and size of periodic and quasi-periodic orbits, using the numerical technique of Poincare surface of sections. Some of the periodic orbits change to quasi-periodic orbits due to the effect of oblateness and vice-versa. The stability of the orbits around Saturn, Titan and both varies with the inclusion of oblateness. The centers of the periodic orbits around Titan move towards Saturn, whereas those around Saturn move towards Titan. For the orbit around Titan at C=2.9992, x=0.959494, the apocenter becomes pericenter. By incorporating oblateness effect, the orbit around Titan at C=2.99345, x=0.924938 is captured by Saturn, remains in various trajectories around Saturn, and as time progresses it spirals away around both the primaries. 相似文献
10.
Stefano Carletta Mauro Pontani Paolo Teofilatto 《Celestial Mechanics and Dynamical Astronomy》2018,130(7):46
This research aims at ascertaining the existence and characteristics of natural long-term capture orbits around a celestial body of potential interest. The problem is investigated in the dynamical framework of the three-dimensional circular restricted three-body problem. Previous numerical work on two-dimensional trajectories provided numerical evidence of Conley’s theorem, proving that long-term capture orbits are topologically located near trajectories asymptotic to periodic libration point orbits. This work intends to extend the previous investigations to three-dimensional paths. In this dynamical context, several special trajectories exist, such as quasiperiodic orbits. These can be found as special solutions to the linear expansion of the dynamics equations and have already been proven to exist even using the nonlinear equations of motion. The nature of long-term capture orbits is thus investigated in relation to the dynamical conditions that correspond to asymptotic trajectories converging into quasiperiodic orbits. The analysis results in the definition of two parameters characterizing capture condition and the design of a capture strategy, guiding a spacecraft into long-term capture orbits around one of the primaries. Both the results are validated through numerical simulations of the three-dimensional nonlinear dynamics, including fourth-body perturbation, with special focus on the Jupiter–Ganymede system and the Earth–Moon system. 相似文献
11.
This paper studies the relative motion of satellite formation flying in arbitrary elliptical orbits with no perturbation.
The trajectories of the leader and follower satellites are projected onto the celestial sphere. These two projections and
celestial equator intersect each other to form a spherical triangle, in which the vertex angles and arc-distances are used
to describe the relative motion equations. This method is entitled the reference orbital element approach. Here the dimensionless
distance is defined as the ratio of the maximal distance between the leader and follower satellites to the semi-major axis
of the leader satellite. In close formations, this dimensionless distance, as well as some vertex angles and arc-distances
of this spherical triangle, and the orbital element differences are small quantities. A series of order-of-magnitude analyses
about these quantities are conducted. Consequently, the relative motion equations are approximated by expansions truncated
to the second order, i.e. square of the dimensionless distance. In order to study the problem of periodicity of relative motion,
the semi-major axis of the follower is expanded as Taylor series around that of the leader, by regarding relative position
and velocity as small quantities. Using this expansion, it is proved that the periodicity condition derived from Lawden’s
equations is equivalent to the condition that the Taylor series of order one is zero. The first-order relative motion equations,
simplified from the second-order ones, possess the same forms as the periodic solutions of Lawden’s equations. It is presented
that the latter are further first-order approximations to the former; and moreover, compared with the latter more suitable
to research spacecraft rendezvous and docking, the former are more suitable to research relative orbit configurations. The
first-order relative motion equations are expanded as trigonometric series with eccentric anomaly as the angle variable. Except
the terms of order one, the trigonometric series’ amplitudes are geometric series, and corresponding phases are constant both
in the radial and in-track directions. When the trajectory of the in-plane relative motion is similar to an ellipse, a method
to seek this ellipse is presented. The advantage of this method is shown by an example. 相似文献
12.
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
13.
Analytical description of physical librations of saturnian coorbital satellites Janus and Epimetheus
Janus and Epimetheus are famously known for their distinctive horseshoe-shaped orbits resulting from a 1:1 orbital resonance. Every 4 years these two satellites swap their orbits by a few tens of kilometers as a result of their close encounter. Recently Tiscareno et al. (Tiscareno, M.S., Thomas, P.C., Burns, J.A. [2009]. Icarus 204, 254-261) have proposed a model of rotation based on images from the Cassini orbiter. These authors inferred the amplitude of rotational librational motion in longitude at the orbital period by fitting a shape model to Cassini ISS images. By a quasi-periodic approximation of the orbital motion, we describe how the orbital swap impacts the rotation of the satellites. To that purpose, we have developed a formalism based on quasi-periodic series with long- and short-period librations. In this framework, the amplitude of the libration at the orbital period is found proportional to a term accounting for the orbital swap. We checked the analytical quasi-periodic development by performing a numerical simulation and find both results in good agreement. To complete this study, the results obtained for the short-period librations are studied with the help of an adiabatic-like approach. 相似文献
14.
Giancarmine Fasano Marco D’Errico 《Celestial Mechanics and Dynamical Astronomy》2009,105(1-3):113-139
While trajectory design for single satellite Earth observation missions is usually performed by means of analytical and relatively simple models of orbital dynamics including the main perturbations for the considered cases, most literature on formation flying dynamics is devoted to control issues rather than mission design. This work aims at bridging the gap between mission requirements and relative dynamics in multi-platform missions by means of an analytical model that describes relative motion for satellites moving on near circular low Earth orbits. The development is based on the orbital parameters approach and both the cases of close and large formations are taken into account. Secular Earth oblateness effects are included in the derivation. Modeling accuracy, when compared to a nonlinear model with two body and J2 forces, is shown to be of the order of 0.1% of relative coordinates for timescales of hundreds of orbits. An example of formation design is briefly described shaping a two-satellite formation on the basis of geometric requirements for synthetic aperture radar interferometry. 相似文献
15.
George Voyatzis John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):259-271
We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit. 相似文献
16.
17.
This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system.
The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and
Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation
of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for
the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the
dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation
of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood
of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the
vertical (z and ż) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov
family of periodic orbits of the Lagrangian points that goes in the (z, ż) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a
very long time. It is remarkable that these regions seem to persist in the real system.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
19.
Michael E. Hough 《Celestial Mechanics and Dynamical Astronomy》1993,57(4):587-617
A new theory is formulated for the analytic continuation of quasi-periodic orbits from non-linear, periodic modes of a two degree-of-freedom system in rotating coordinates. Formal solutions of the equations of motion are developed in series expansions, with the non-linear modes as reference solutions. Conditions are determined on the existence and boundedness of the time-dependent coefficients of the velocity field. Boundedness properties of the orbits are specified by an orbit function derived from the Jacobi integral. 相似文献
20.
Analysis of periodic orbits in the Saturn-Titan system using the method of Poincare section surfaces
We explore the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system
in the framework of planar circular restricted three-body problem. The location, nature and size of periodic and quasi-periodic
orbits are studied using the numerical technique of Poincare surface of sections. The maximum amplitude of oscillations about
the periodic orbits is determined and is used as a parameter to measure the degree of stability in the phase space for such
orbits. It is found that the orbits around Saturn remain around it and their stability increases with the increase in the
value of Jacobi constant C. The orbits around Titan move towards it with the increase in C. At C=3.1, the pericenter and apocenter are 358.2 and 358.5 km, respectively. No periodic or quasi-periodic orbits could be found
by the present method around the collinear Lagrangian point L
1 (0.9569373834…). 相似文献