Optimal transfers between unstable periodic orbits using invariant manifolds

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.     

Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth–Moon Missions

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 \(L_2\) .     

Optimal low-thrust transfers between libration point orbits

Over the past three decades, ballistic and impulsive trajectories between libration point orbits (LPOs) in the Sun–Earth–Moon system have been investigated to a large extent. It is known that coupling invariant manifolds of LPOs of two different circular restricted three-body problems (i.e., the Sun–Earth and the Earth–Moon systems) can lead to significant mass savings in specific transfers, such as from a low Earth orbit to the Moon’s vicinity. Previous investigations on this issue mainly considered the use of impulsive maneuvers along the trajectory. Here we investigate the dynamical effects of replacing impulsive ΔV’s with low-thrust trajectory arcs to connect LPOs using invariant manifold dynamics. Our investigation shows that the use of low-thrust propulsion in a particular phase of the transfer and the adoption of a more realistic Sun–Earth–Moon four-body model can provide better and more propellant-efficient solution. For this purpose, methods have been developed to compute the invariant tori and their manifolds in this dynamical model.     

Lunar capture trajectories and homoclinic connections through isomorphic mapping

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.     

A two-level perturbation method for connecting unstable periodic orbits with low fuel cost and short time of flight: application to a lunar observation mission

The proposed method connects two unstable periodic orbits by employing trajectories of their associated invariant manifolds that are perturbed in two levels. A first level of velocity perturbations is applied on the trajectories of the discretized manifolds at the points where they approach the nominal unstable periodic orbit in order to accelerate them. A second level of structured velocity perturbations is applied to trajectories that have already been subjected to first level perturbations in order to approximately meet the necessary conditions for a low \(\varDelta \text {V}\) transfer. Due to this two-level perturbation approach, the number of the trajectories obtained is significantly larger compared with approaches that employ traditional invariant manifolds. For this reason, the problem of connecting two unstable periodic orbits through perturbed trajectories of their manifolds is transformed into an equivalent discrete optimization problem that is solved with a very low computational complexity algorithm that is proposed in this paper. Finally, the method is applied to a lunar observation mission of practical interest and is found to perform considerably better in terms of \(\varDelta \text {V}\) cost and time of flight when compared with previous techniques applied to the same project.     

Earth–Mars halo to halo low thrust manifold transfers

Application of low thrust propulsion to interconnect ballistic trajectories on invariant manifolds associated with multiple circular restricted three body systems has been investigated. Sun-planet three body models have been coupled to compute the two ballistic trajectories, where electric propulsion is used to interconnect these trajectories as no direct intersection in the Poincarè sections exists. The ability of a low thrust to provide the energy change required to transit the spacecraft between two systems has been assessed for some Earth to Mars transfers. The approach followed consists in a planetary escape on the unstable manifold starting from a periodic orbit around one of the two collinear libration points near the secondary body. Following the planetary escape and the subsequent coasting phase, the electric thruster is activated and executes an ad-hoc thrusting phase. The complete transfer design, composed of the three discussed phases, and possible applications to Earth–Mars missions is developed where the results are outlined in this paper.     

Impulsive time-free transfers between halo orbits

A methodology is developed to design optimal time-free impulsive transfers between three-dimensional halo orbits in the vicinity of the interior L ₁ libration point of the Sun-Earth/Moon barycenter system. The transfer trajectories are optimal in the sense that the total characteristic velocity required to implement the transfer exhibits a local minimum. Criteria are established whereby the implementation of a coast in the initial orbit, a coast in the final orbit, or dual coasts accomplishes a reduction in fuel expenditure. The optimality of a reference two-impulse transfer can be determined by examining the slope at the endpoints of a plot of the magnitude of the primer vector on the reference trajectory. If the initial and final slopes of the primer magnitude are zero, the transfer trajectory is optimal; otherwise, the execution of coasts is warranted. The optimal time of flight on the time-free transfer, and consequently, the departure and arrival locations on the halo orbits are determined by the unconstrained minimization of a function of two variables using a multivariable search technique. Results indicate that the cost can be substantially diminished by the allowance for coasts in the initial and final libration-point orbits.An earlier version was presented as Paper AIAA 92-4580 at the AIAA/AAS Astrodynamics Conference, Hilton Head Island, SC, U.S.A., August 10–12, 1992.     

Search for and study of nearly periodic orbits in the plane problem of three equal-mass bodies

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.     

Orbits and manifolds near the equilibrium points around a rotating asteroid

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.     

Instability and Diffusion in the Elliptic Restricted Three-body Problem

The importance of the stability characteristics of the planar elliptic restricted three-body problem is that they offer insight about the general dynamical mechanisms causing instability in celestial mechanics. To analyze these concerns, elliptic–elliptic and hyperbolic–elliptic resonance orbits (periodic solutions with lower period) are numerically discovered by use of Newton's differential correction method. We find indications of stability for the elliptic–elliptic resonance orbits because slightly perturbed orbits define a corresponding two-dimensional invariant manifold on the Poincaré surface-section. For the resonance orbit of the hyperbolic–elliptic type, we show numerically that its stable and unstable manifolds intersect transversally in phase-space to induce instability. Then, we find indications that there are orbits which jump from one resonance zone to the next before escaping to infinity. This phenomenon is related to the so-called Arnold diffusion. This revised version was published online in August 2006 with corrections to the Cover Date.     

Combined low-thrust propulsion and invariant manifold trajectories to capture NEOs in the Sun–Earth circular restricted three-body problem

In this paper, a method to capture near-Earth objects (NEOs) incorporating low-thrust propulsion into the invariant manifolds technique is investigated. Assuming that a tugboat-spacecraft is in a rendez-vous condition with the candidate asteroid, the aim is to take the joint spacecraft-asteroid system to a selected periodic orbit of the Sun–Earth restricted three-body system: the orbit can be either a libration point periodic orbit (LPO) or a distant prograde periodic orbit (DPO) around the Earth. In detail, low-thrust propulsion is used to bring the joint spacecraft-asteroid system from the initial condition to a point belonging to the stable manifold associated to the final periodic orbit: from here onward, thanks to the intrinsic dynamics of the physical model adopted, the flight is purely ballistic. Dedicated guided and capture sets are introduced to exploit the combined use of low-thrust propulsion with stable manifolds trajectories, aiming at defining feasible first guess solutions. Then, an optimal control problem is formulated to refine and improve them. This approach enables a new class of missions, whose solutions are not obtainable neither through the patched-conics method nor through the classic invariant manifolds technique.     

Escape regions of a quartic potential

We investigate the escape regions of a quartic potential and the main types of irregular periodic orbits. Because of the symmetry of the model the zero velocity curve consists of four summetric arcs forming four open channels around the lines y = ± x through which an orbit can escape. Four unstable Lyapunov periodic orbits bridge these openings.We have found an infinite sequence of families of periodic orbits which is the outer boundary of one of the escape regions and several infinite sequences of periodic orbits inside this region that tend to homoclinic and heteroclinic orbits. Some of these sequences of periodic orbits tend to homoclinic orbits starting perpendicularly and ending asymptotically at the x-axis. The other sequences tend to heteroclinic orbits which intersect the x-axis perpendicularly for x > 0 and make infinite oscillations almost parallel to each of the two Lyapunov orbits which correspond to x > 0 or x < 0.     

Strategies for plane change of Earth orbits using lunar gravity and derived trajectories of family G

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.     

Motion close to the Hopf Bifurcation Of the vertical family of periodic orbits of L4

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) μ_R (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.     

Asymptotic orbits in the (<Emphasis Type="Italic">N</Emphasis>+1)-body ring problem

In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m ₀=β m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov periodic orbits, which emanate from the collinear equilibrium points L ₁ and L ₂. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points. All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented.     

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L ₁and L ₂, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L ₁and L ₂is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L ₁and the other around L ₂) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.     

The Restricted Hill Four-Body Problem with Applications to the Earth–Moon–Sun System

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.     

Study of the transfer between libration point orbits and lunar orbits in Earth–Moon system

This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio.     

Using FLI maps for preliminary spacecraft trajectory design in multi-body environments

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.     

Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem

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.