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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.
Martin Lara 《Celestial Mechanics and Dynamical Astronomy》2017,127(3):285-300
This study analyzes a recently discovered class of exterior transfers to the Moon. These transfers terminate in retrograde ballistic capture orbits, i.e., orbits with negative Keplerian energy and angular momentum with respect to the Moon. Yet, their Jacobi constant is relatively low, for which no forbidden regions exist, and the trajectories do not appear to mimic the dynamics of the invariant manifolds of the Lagrange points. This paper shows that these orbits shadow instead lunar collision orbits. We investigate the dynamics of singular, lunar collision orbits in the Earth–Moon planar circular restricted three-body problem, and reveal their rich phase space structure in the medium-energy regime, where invariant manifolds of the Lagrange point orbits break up. We show that lunar retrograde ballistic capture trajectories lie inside the tube structure of collision orbits. We also develop a method to compute medium-energy transfers by patching together orbits inside the collision tube and those whose apogees are located in the appropriate quadrant in the Sun–Earth system. The method yields the novel family of transfers as well as those ending in direct capture orbits, under particular energetic and geometrical conditions. 相似文献
3.
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
4.
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
5.
We study two and three-dimensional resonant periodic orbits, usingthe model of the restricted three-body problem with the Sun andNeptune as primaries. The position and the stability character ofthe periodic orbits determine the structure of the phase space andthis will provide useful information on the stability and longterm evolution of trans-Neptunian objects. The circular planarmodel is used as the starting point. Families of periodic orbitsare computed at the exterior resonances 1/2, 2/3 and 3/4 withNeptune and these are used as a guide to select the energy levelsfor the computation of the Poincaré maps, so that all basicresonances are included in the study. Using the circular planarmodel as the basic model, we extend our study to more realisticmodels by considering an elliptic orbit of Neptune and introducingthe inclination of the orbit. Families of symmetric periodicorbits of the planar elliptic restricted three-body problem andthe three-dimensional problem are found. All these orbitsbifurcate from the families of periodic orbits of the planarcircular problem. The stability of all orbits is studied. Althoughthe resonant structure in the circular problem is similar for allresonances, the situation changes if the eccentricity of Neptuneor the inclination of the orbit is taken into account. All theseresults are combined to explain why in some resonances there aremany bodies and other resonances are empty. 相似文献
6.
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
7.
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
8.
Kathryn E. Davis Rodney L. Anderson Daniel J. Scheeres George H. Born 《Celestial Mechanics and Dynamical Astronomy》2011,109(3):241-264
This paper presents a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant
manifolds. The transfers constructed in this method asymptotically depart the initial orbit on a trajectory contained within
the unstable manifold of the initial orbit and later, asymptotically arrive at the final orbit on a trajectory contained within
the stable manifold of the final orbit. Primer vector theory is applied to a transfer to determine the optimal maneuvers required
to create the bridging trajectory that connects the unstable and stable manifold trajectories. Transfers are constructed between
unstable periodic orbits in the Sun–Earth, Earth–Moon, and Jupiter-Europa three-body systems. Multiple solutions are found
between the same initial and final orbits, where certain solutions retrace interior portions of the trajectory. All transfers
created satisfy the conditions for optimality. The costs of transfers constructed using manifolds are compared to the costs
of transfers constructed without the use of manifolds. In all cases, the total cost of the transfer is significantly lower
when invariant manifolds are used in the transfer construction. In many cases, the transfers that employ invariant manifolds
are three times more efficient, in terms of fuel expenditure, than the transfer that do not. The decrease in transfer cost
is accompanied by an increase in transfer time of flight. 相似文献
9.
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
10.
Transit orbits are defined as the trajectories that can pass through the neck region of the zero velocity surface in the circular restricted three-body problem (CR3BP). The low-energy transfers in the CR3BP or between two CR3BPs are always through the instrumentality of the transit orbits. In this paper, the distribution of the transit orbits in the six-dimensional phase space is explored by using numerical methods. The necessary and sufficient condition of transition is introduced, which defines the distribution of the transit orbits by using the manifolds of the vertical and horizontal Lyapunov orbits and the transit cones. The relationship between the manifolds of the libration point orbits and the boundary of the transit orbits is discovered. By using this relationship, a fast algorithm for detecting the boundary of the transit orbits is developed. Moreover, this boundary is parametrized by using Fourier series, which makes easy to use the conclusions of this paper in future trajectory optimization and mission design. All the analyses in this paper are based on the Sun?CEarth CR3BP, but the methods introduced here can be extended to any CR3BPs. 相似文献
11.
M. Michalodimitrakis 《Astrophysics and Space Science》1979,65(2):459-475
In this paper we present a two-parametric family of symmetric periodic orbits of the three-dimensional general three-body problem, found numerically by continuation of a vertical critical orbit of the circular restricted three-body problem. The periodic orbits refer to a suitably defined rotating frame of reference. 相似文献
12.
K. E. Papadakis 《Astrophysics and Space Science》2006,302(1-4):67-82
We study numerically the asymptotic homoclinic and heteroclinic orbits around the hyperbolic Lyapunov periodic orbits which
emanate from Euler's critical points L
1 and L
2, in the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated
to these Lyapunov orbits, are also presented. Poincaré surface of sections of these manifolds on appropriate planes and several
homoclinic and heteroclinic orbits for the gravitational case as well as for varying radiation factor q
1, are displayed. Homoclinic-homoclinic and homoclinic-heteroclinic-homoclinic chains which link the interior with the exterior
Hill's regions, are illustrated. We adopt the Sun-Jupiter system and assume that only the larger primary radiates. It is found
that for small deviations of its value from the gravitational case (q
1 = 1), the radiation pressure exerts a significant impact on the Hill's regions and on these asymptotic orbits. 相似文献
13.
Rodney L. Anderson Jeffrey S. Parker 《Celestial Mechanics and Dynamical Astronomy》2013,115(3):311-331
In this study, transfer trajectories from the Earth to the Moon that encounter the Moon at various flight path angles are examined, and lunar approach trajectories are compared to the invariant manifolds of selected unstable orbits in the circular restricted three-body problem. Previous work focused on lunar impact and landing trajectories encountering the Moon normal to the surface, and this research extends the problem with different flight path angles in three dimensions. The lunar landing geometry for a range of Jacobi constants is computed, and approaches to the Moon via invariant manifolds from unstable orbits are analyzed for different energy levels. 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.
16.
Numerical orbit integrations have been conducted to characterize the types of trajectories in the one-dimensional Newtonian
three-body problem with equal masses and negative energy. Essentially three different types of motions were found to exist.
They may be classified according to the duration of the bound three-body state. There are zero-lifetime predictable trajectories,
finite lifetime apparently chaotic orbits, and infinite lifetime quasi-periodic motions. The quasi-periodic orbits are confined
to the neighbourhood of Schubart's stable periodic orbit. For all other trajectories the final state is of the type binary
+ single particle in both directions of time. The boundaries of the different orbit-type regions seem to be sharp. We present
statistical results for the binding energies and for the duration of the bound three-body state. Properties of individual
orbits are also summarized in the form of various graphical maps in a two-dimensional grid of parameters defining the orbit.
Supported by the Academy of Finland. 相似文献
17.
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
18.
This work studies a special type of cislunar periodic orbits in the circular restricted three-body problem called resonance transition periodic orbits, which switch between different resonances and revolve about the secondary with multiple loops during one period. In the practical computation, families of multiple periodic orbits are identified first, and then the invariant manifolds emanating from the unstable multiple periodic orbits are taken to generate resonant homoclinic connections, which are used to determine the initial guesses for computing the desired periodic orbits by means of multiple-shooting scheme. The obtained periodic orbits have potential applications for the missions requiring long-term continuous observation of the secondary and tour missions in a multi-body environment. 相似文献
19.
Asymptotic motion near the collinear equilibrium points of the photogravitational restricted three-body problem is considered.
In particular, non-symmetric homoclinic solutions are numerically explored. These orbits are connected with periodic ones.
We have computed numerically the families containing these orbits and have found that they terminate at both ends by asymptotically
approaching simple periodic solutions belonging to the Lyapunov family emanating from L3. 相似文献
20.
C. S. Arnot C. R. McInnes R. J. McKay M. Macdonald J. Biggs 《Celestial Mechanics and Dynamical Astronomy》2018,130(2):12
This paper presents rich new families of relative orbits for spacecraft formation flight generated through the application of continuous thrust with only minimal intervention into the dynamics of the problem. Such simplicity facilitates implementation for small, low-cost spacecraft with only position state feedback, and yet permits interesting and novel relative orbits in both two- and three-body systems with potential future applications in space-based interferometry, hyperspectral sensing, and on-orbit inspection. Position feedback is used to modify the natural frequencies of the linearised relative dynamics through direct manipulation of the system eigenvalues, producing new families of stable relative orbits. Specifically, in the Hill–Clohessy–Wiltshire frame, simple adaptations of the linearised dynamics are used to produce a circular relative orbit, frequency-modulated out-of-plane motion, and a novel doubly periodic cylindrical relative trajectory for the purposes of on-orbit inspection. Within the circular restricted three-body problem, a similar minimal approach with position feedback is used to generate new families of stable, frequency-modulated relative orbits in the vicinity of a Lagrange point, culminating in the derivation of the gain requirements for synchronisation of the in-plane and out-of-plane frequencies to yield a singly periodic tilted elliptical relative orbit with potential use as a Lunar far-side communications relay. The \(\Delta v\) requirements for the cylindrical relative orbit and singly periodic Lagrange point orbit are analysed, and it is shown that these requirements are modest and feasible for existing low-thrust propulsion technology. 相似文献