Theory of Orbits: The Restricted Problem of Three Bodies
- ... To facilitate numerical integration, the dimensional values are nondimensionalized by characteristic quantities such that the distance between P 1 and P 2 is unity, the mean motion of the two primaries is unity, and the masses of each body range from zero to one. 10 The spacecraft is located relative to the system barycenter in the rotating frame via ...... In the ballistic CR3BP, five equilibrium solutions exist: the L 1 , L 2 , and L 3 points located on the rotatingˆxrotatingˆ rotatingˆx-axis, and the L 4 and L 5 points off-axis in the xy-plane. 10 When a low- thrust force is included in the model, the equilibrium solutions are perturbed from the ballistic solutions, as depicted in Figure 2 for the planar Earth-Moon CR3BP-LT. In this example, the low- (a) Low-thrust equilibria throughout the system (b) A close-up view of the E1 and E2 low-thrust equi- libria (colored) near the natural L1 and L2 points Figure 2. Low-thrust equilibrium solutions (colored by α) in the Earth-Moon CR3BP-LT for a lt = 7e-2 and α ∈ [−π, π]; the natural equilibrium solutions are included as black asterisks. ...... Recall that the center mode associated with the L 1 and L 2 solutions offers an initial approximation for the well-known Lyapunov orbits in the natural CR3BP. 10 Similarly, periodic solutions in the CR3BP-LT are predicted by the existence of the center mode associated with the E 1 and E 2 equilibria. 8, 13 The saddle mode approximates unstable and stable manifolds that asymptotically approach the equilibrium point in forward and reverse time, respectively. ...Path planning in the circular restricted 3-body problem (CR3BP) is frequently guided by the forbidden regions and manifold arcs. However, when low-thrust is employed to modify the spacecraft trajectory, these dynamical structures pulsate with the varying Hamiltonian value. In a combined CR3BP, low-thrust (CR3BP- LT) model, an additional low-thrust Hamiltonian is available that remains constant along low-thrust arcs. Accordingly, the analogous low-thrust forbidden regions and manifolds are static and are useful guides for low-thrust trajectory design. Strategies leveraging these structures and other insights from the CR3BP-LT are explored to construct transit and capture itineraries.
- ... The equations of motion of the infinitesimal mass m are we choose the sum of the masses of the primaries to be one so that if m 2 = , m 1 = 1-, where is the ratio of the mass of the smaller primary to the total mass of the primaries and 0 < ½.The distance between the primaries is taken equal to one and the unit of time is so chosen as to make the gravitational constant G is unity. We have the origin as the barycentre of the masses m 1 and m 2 as defined by Szebehely (1967b). The equations of motion in the dimensionless synodic coordinate system (Oxyz) becomes ...... If the bigger primary is not triaxial ( 1 = 2 = 0), the smaller primary not radiating (q= 1) and there are no perturbations in the coriolis and centrifugal forces, then the equations of motion reduce to that of classical problem (Szebehely, 1967b). In the absence of triaxiality and radiation, the equation of motion coincides with those obtained by Bhatnagar and Hallan (1979). ...... In the absence of perturbation potentials, triaxiality of the bigger primary and radiation force of the smaller primary, the critical value of the mass parameter reduces to ... 038520 . 0 27 Szebehely, 1967b) which is mass ratio for the classical restricted problem. But in the absence of only perturbation potential ( p = 0) the critical mass value c agrees with that of . ...This study is aimed at studying the effects of small perturbations in the coriolis () and th centrifugal (΄) forces, the triaxiality (1, 2) of the bigger primary and the radiation (q) pressure force of the smaller primary on the stability of libration point in the Restricted Three-Body Problem(RTBP) in particular to study the effect of perturbations in the coriolis and the centrifugal forces on the of the libration points in the restricted three body problem when the bigger primary is a triaxial rigid body and the smaller primary a source of radiation. The equations of motion of the restricted problem under influence of the perturbations in the coriolis and the centrifugal forces, triaxiality and radiation were established. These equations of motion are found to be affected by the aforesaid parameters. They generalize the classical equations of motion of restricted problem and those obtained by others. Five libration points were obtained three collinear points (𝐿1, 𝐿2, 𝐿3) and two triangular points (𝐿4, 𝐿5). The libration points were found to be affected by the small perturbation for the range 0 ≤ < c. The critical mass value c is affected by all the aforesaid parameters in the coriolis and centrifugal forces, the triaxial nature of the bigger primary and the radiation pressure force of the smaller primary. This generalizes the equation of orbits of the classical restricted three body problem and those obtained by others.
- ... Consequently, one can formulate a problem of studying the motion of particle P 3 of negligible mass in the gravitational field created by particles P 0 , P 1 of masses m 0 , m 1 , respectively, moving on circular orbits about their common center of mass according to the corresponding solution of the two-body problem. Such a model was first proposed by L. Euler and is known as the circular restricted three-body problem (see [22]). As in case of a general three-body problem, there exist five exact particular solutions of the differential equations of motion determining the equilibrium positions of particle P 3 in the rotating coordinate system; these equilibria are called the points of libration L j (j = 1, 2, ..., 5). ...... The libration points are of great interest for applications and so their stability was a subject of many papers during the past two hundred years. As a result it was proven that three points L 1 , L 2 , L 3 situated at the line P 0 P 1 (collinear equilibrium positions) are unstable while the libration points L 4 , L 5 (triangular equilibrium positions) may be stable if the mass ratio µ 1 = m 1 /m 0 is sufficiently small (see [12,22]). ...... In such a system the particles P 1 , P 2 are immovable and located on the same straight line which may be considered as the Ox axis. Without loss of generality, one can consider also that the dimensionless x-coordinates of particles P 1 , P 2 are equal to 1 and a, respectively, where the variable a is defined by the equation (see [10,17,22]) ...We discuss here the problem of solving the system of two nonlinear algebraic equations determining the relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The system contains two parameters $\mu_1$, $\mu_2$ and all its solutions coincide with the corresponding solutions in the three-body problem if one of the parameters equals to zero. For small values of one parameter the solutions are found in the form of power series in terms of this parameter, and they are used for separation of different solutions and choosing the starting point in the numerical procedure for the search of equilibria. Combining symbolic and numerical computation, we found all the equilibrium positions and proved that there are 18 different equilibrium configurations of the system for any reasonable values of the two system parameters $\mu_1$, $\mu_2$. All relevant symbolic and numerical calculations are performed with the aid of the computer algebra system Wolfram Mathematica.
- ... The orbit of the Jupiter around the Sun is a fixed ellipse and the Trojan asteroids are influenced by the gravitational attraction of the Sun and Jupiter. The stability of such systems (ER3BP) moving in elliptic orbits was investigated by many authors, [1][2][3][4][5][6][7][8] and many others. ...... The differential equations of motion of the infinitesimal mass in the elliptic restricted three body problem under radiating and triaxial primaries in the barycentric, pulsating and rotating, non-dimensional coordinates are derived in [19] and given in equation (1). The notations in principle are taken from [4] with some minor modifications in the notation being done for adapting to the present problem, presented as : ...... The instability can also be seen from the graph( figure:2) (iii) For í µí°¿ 3 , the motion appears to be stable for some values of í µí»¿, í µí¼ 1 and í µí¼ 2 because the values of k < 1 as well as í µí¼ 1,2 2 < 0 . As, the value of í µí¼ 1 and í µí¼ 2 , increases the system becomes unstable, for a fixed value of í µí»¿ , the roots of the system í µí¼ 1,2 2 < 0. That is, the roots will be imaginary implying the stability of the system which is evident from the table: [4] and also from the graph( figure:3) When í µí¼ 1 and í µí¼ 2 are negligibly small, the locations and stability of our results are in confirmation with [16] and [17]. The collinear point í µí°¿ 3 was found to be stable for the value of radiation pressure í µí»½ ≥ 0.115 in [16] and [17]. ...The present paper studies the motion of an infinitesimal mass around a stellar primary and triaxial secondary moving around each other in the elliptic orbits about their common center of mass in the neighbourhood of Collinear equilibrium points. The location and stability of the collinear points are found to be affected by the radiation pressure and triaxility parameters. The nature of stability, however, remains unchanged. The collinear points L1 and L2 are unstable in the Lyapunov sense. But, the collinear point L3 shows a stable behavior for small values of radiation and triaxiality parameters.
- ... z) zero to one. 19 The spacecraft is located relative to the system barycenter in the rotating frame via the vector r = {x y z} T . ...... Bounds on the spacecraft motion in the natural CR3BP, termed forbidden regions, are linked to the instantaneous value of H nat along a trajectory, thus, information about the evolution of H nat provided by an energy plane is useful to plan for desirable configurations of the forbidden regions. 19 The H nat values associated with the natural equilibrium solutions are significant as they represent critical H nat values at which the forbidden regions shrink (or grow) to permit (or restrict) access to specific locations in the rotating frame. For example, for H nat values slightly higher than the H nat (L 1 ) value, the forbidden regions include a narrow neck near the L 1 point, i.e., a "gateway," through which trajectories may pass to transit between the P 1 and P 2 regions. ...... Global invariant manifolds are constructed by transitioning the linear results to the nonlinear model. 19 Manipulations of the low-thrust acceleration vector directly influence the number and location of equilibrium solutions in the CR3BP-LT, which subsequently affects the existence and characteristics of various nearby dynamical structures. Accordingly, the equilibrium solutions in the CR3BP-LT are relevant to low-thrust mission applications, particularly as the equilibria locations evolve relative to the familiar CR3BP equilibrium points. ...A key challenge in low-thrust trajectory design is generating preliminary solutions that simultaneously specify the spacecraft position and velocity vectors, as well as the thrust history. To mitigate this difficulty, dynamical structures within a combined low-thrust circular restricted 3-body problem (CR3BP) are investigated as candidate solutions to seed initial low-thrust trajectory designs. The addition of low-thrust to the CR3BP modifies the locations and stability of the equilibria, offering novel geometries for mission applications. Transfers between these novel equilibria are constructed by leveraging the associated stable and unstable mani-folds and insights from the low-thrust CR3BP.
- ... A growing interest of the space scientific community for trajectories toward, around and from Lagrangian points has been registered in recent years. In particular, the three-body problem (Szebehely, 1967) is one of the most studied models not only in celestial mechanics, but also in mathematics. For the early first solar system exploration missions (like Voyager), a patched conics model was satisfactory to compute the trajectory. ...... The collinear points L 1 , L 2, and L 3 are on the line connecting the two primaries, while L 4 and L 5 are equilateral points. In this paper, the distance from L i , to the smallest primary, is named γ i (Szebehely, 1967). According to the literature (Szebehely, 1967;Farquhar, 1972Farquhar, , 1973Whitley and Martinez, 2016), several families of orbits around them exist, usually designated as: Lissajous orbits, Horizontal Lyapunov orbits, Vertical Lyapunov orbits, Halo orbits (including Near Rectilinear Halo Orbits) or Distant Retrograde Orbits. ...... In this paper, the distance from L i , to the smallest primary, is named γ i (Szebehely, 1967). According to the literature (Szebehely, 1967;Farquhar, 1972Farquhar, , 1973Whitley and Martinez, 2016), several families of orbits around them exist, usually designated as: Lissajous orbits, Horizontal Lyapunov orbits, Vertical Lyapunov orbits, Halo orbits (including Near Rectilinear Halo Orbits) or Distant Retrograde Orbits. This paper mainly focuses on Halo orbits, which are three-dimensional and periodic with the same in-and out-of-plane oscillation and NRHO, which are particular case of Halo orbits, with a close passage over a lunar pole. ...In the context of Human Spaceflight exploration mission scenario, with the Lunar Orbital Platform- Gateway (LOP-G) orbiting about Earth-Moon Lagrangian Point (EML), Rendezvous and Docking (RVD) operational activities are mandatory and critical for the deployment and utilization of the LOP-G (station assembly, crew rotations, cargo delivery, lunar sample return). There is extensive experience with RVD in the two-body problem: in Low Earth Orbit (LEO) to various space stations, or around quasi-circular Low Lunar Orbits (LLO), the latter by Apollo by means of manual RVD. However, the RVD problem in non-Keplerian environments has rarely been addressed and no RVD has been performed to this date in the vicinity of Lagrangian points (LP) where Keplerian dynamics are no longer applicable. Dynamics in such regions are more complex, but multi-body dynamics also come with strong advantages that need to be further researched by the work proposed here. The aim of this paper is to present methods and results of investigations conducted to first set up strategies for far and close rendezvous between a target (the LOP-G, for example) and a chaser (cargo, crew vehicle, ascent and descent vehicle, station modules, etc.) depending on target and chaser orbit. Semi-analytical tools have been developed to compute and model families of orbits about the Lagrangian points in the Circular Restricted Three Body Problem (CR3BP) like NRHO, DRO, Lyapunov, Halo and Lissajous orbits. As far as close rendezvous is concerned, implementation of different linear and non-linear models used to describe cis-lunar relative motion will be discussed and compared, in particular for NRHO and DRO.
- ... The three-body problem has been extensively studied during the past two centuries and attracted many authors from Poincare till now. Szebehely made an extensive treatment of the problem [1]. More recently, numerical techniques have been used to generate solutions. ...... The three collinear points L1, L2 and L3, can be found from Equation (13) [1]. The libration point L1 lies between masses m1 and m2 we can calculate it from nonlinear equation. ...... The stability of motion near an equilibrium point in the nonlinear system can be obtained by linearizing and producing variationally equations relative to the equilibrium solutions [1]. Thus, a limited investigation of the motion of spacecraft in the vicinity of any libration point can be accomplished with linear analysis. ...
- ... The circular restricted three-body problem (for mass points) is the three-body problem considered in the case where one of these bodies has a mass, which is so small that we can neglect the influence of this mass on circular orbits of two other bodies [1,2]. Nevertheless, until now, the three-body problem of this kind is still attractive for many investigators who can suggest a lot of interesting applications of this model [1,[3][4][5][6][7][8][9]. ...... The circular restricted three-body problem (for mass points) is the three-body problem considered in the case where one of these bodies has a mass, which is so small that we can neglect the influence of this mass on circular orbits of two other bodies [1,2]. Nevertheless, until now, the three-body problem of this kind is still attractive for many investigators who can suggest a lot of interesting applications of this model [1,[3][4][5][6][7][8][9]. ...... ???? ? ????? ???? are schematically shown in Figs. 1 and 2. Of course, their structure depends on |?| (for details, see [1]). In these figures, the axis ? ...In this paper, we consider the existence conditions of bounded motions of an infinitesimally small particle in the spatial circular restricted three-body problem. In particular, by using the Jacobi integral that corresponds to the inertial coordinate system, we prove a theorem on boundedness of motion that enables us to supplement the Hill approach in the study of motion of an infinitesimally small particle.
- ... In order to justify applicability of the framework we apply it to the Circular Restricted Three Body Problem (CR3BP) [44]. We give a computer-assisted proof of the existence of wide branches of so-called halo orbits bifurcating from the families of planar Lyapunov orbits around L 1,2,3 libration points. ...... Denote by µ the relative mass of the smaller primary. In a rotating coordinate system centred at the common mass centre of two big primaries, the dynamics of the small particle is governed by the following system of second-order differential equations [26,44] ...... They are of saddle-centre type for the PCR3BP. It is well known [44], that for all µ ∈ (0, 1) there exists a family of R-symmetric periodic orbits, called planar Lyapunov orbits (PLO), that surround these libration points -see also Figure 5. In [11,12] the existence of Lyapunov orbits around L 1 and L 2 libration points for selected mass parameters has been proved in an explicit domain. ...We propose a general framework for computer-assisted verification of the presence of symmetry breaking, period-tupling and touch-and-go bifurcations of symmetric periodic orbits for reversible maps. The framework is then adopted to Poincaré maps in reversible autonomous Hamil-tonian systems. In order to justify the applicability of the method, we study bifurcations of halo orbits in the Circular Restricted Three Body Problem. We give a computer-assisted proof of the existence of wide branches of halo orbits bifurcating from L 1,2,3-Lyapunov families and for wide range of mass parameter. For two physically relevant mass parameters we prove, that halo orbits undergo multiple period doubling, quadrupling and third-order touch-and-go bifurcations.
- ... A framework with a good fit to model the dynamics of multi-body en- vironments in the Solar System is the circular restricted tree-body problem (CR3BP), where only two celestial bodies, a primary and a secondary, are considered at a time, and they are assumed to be in circular orbits with re- spect to their barycenter. 5 The CR3BP approximates well the gravitational environment when the dominate bodies are the Sun and a planet or a planet and a moon. The application of dynamical system theory to the CR3BP has allowed to exploit the chaotic nature of low-energy trajectories in a methodical manner. ...... As a brief review, the motion of a non-thrusting spacecraft is governed by the gravity of two bodies B1 and B2 moving in circular orbits around their barycenter. 5 Using dimensionless units in a rotating, barycentric frame, the equations of motion of the spacecraft with position coordinates x, y, and z, are given by ...The implementation of low-energy trajectories has opened a new paradigm in trajectory design, where, at the cost of long transfer times, low-fuel consumption missions opened a new world of routes for space exploration. Low-cost Lunar missions and the exploration of the gas giant planetary satellite systems are one of the main applications of these fuel-efficient trajectories. However, the flexibility in applications that low-energy trajectories provide comes at the cost of high complexity in the design. In this dissertation a systematic strategy to simplify the design process of low-energy trajectories is developed. The method, called patched periodic orbits (PPO) is analogous to the patched conics model in high energy regimes, where conic segments are used as building blocks to give rise to full complex trajectories. In the PPO model, the building blocks are precomputed three- body periodic orbits that are patched together to build transfer mechanisms in multi-body environments. To support the PPO model, a broad database of axisymmetric three-body periodic orbits for planets and main planetary satellites in the Solar System have been generated and provided online. The periodic orbit search is performed over 24 pairs of bodies that are well approximated by the circular restricted three-body problem, resulting in approximately 3 million periodic solutions. The database contains a new set of periodic solutions that approximate heteroclinic connections between other pairs of periodic orbits. These connecting orbits provide free escape/capture mechanisms as well as natural transfers between libration point orbits, among others. In order to efficiently converge the highly sensitive solutions the database is generated using a multiple grid search strategy and a robust differential corrector with a full second-order trust region method Examples of point design applications for several different challenging trajectory problems are introduced, including transfers between the Galilean moons, alternative endgames of a Europa mission, and low-cost Earth-Moon transfers with ballistic lunar captures. Additionally, a strategy to compute landing trajectories at Europa with a broad surface coverage is presented. The strategy uses lissajous segments as staging orbits that allow to decouple the landing phase with the Europa approach. Combining the PPO model and the landing strategy, an end-to-end solution that connects the landing phase, the staging orbit, and a Ganymede-Europa moon tour is presented.
- ... [2][3][4] The CR3BP model has five equilibrium points, all of them on the ecliptic plane (z = 0). 5 When the SRP ac- celeration is added to the model the equilibrium points subsist but their position varies with the parameters of the sail. 8 In this paper we compute two families of equilib- rium points (SL 1 and SL 2 ) for different values of α, δ, and the lightness parameter. ...... As is usual in the CR3BP 5 we use a synodic reference 69th International Astronautical Congress, Bremen, Germany. Copyright 2018 by the authors. ...This paper investigates propellant-free transfers between lissajous libration point orbits using solar sailing. The dynamical model used is the 3-dimensional Sun-Earth restricted three-body problem, including solar radiation pressure (SRP) and the maneuvres under consideration are controlled by the change in the spacecraft reflectivity or by changing the attidude of the sail with respect to the Sun. The study is based on a careful analysis of the geometry of the phase space of the linearised equations around the equilibrium points and, as a matter of fact contrary to the classical impulsive maneuvers, in the geometrical approach, solar sail maneuvers in the libration zone can be understood as jumps in position instead of in velocity.
- ... As mentioned in Section III, the two trans- fers will first be optimized separately, resulting in two time-optimal transfers with non-constant R(α, δ, t), of which the numerical results can be found in Tab. 5 for β = 0.05. Note that the departure and arrival at the halo orbits are not yet according to Eq. [24]. These resulting transfers are subsequently used as ini- tial guess for one larger optimal control problem with six phases, where also the relative timing of the two transfers can be optimized and Eq. ...... These resulting transfers are subsequently used as ini- tial guess for one larger optimal control problem with six phases, where also the relative timing of the two transfers can be optimized and Eq. [24] will be satis- Fig. 14 are obtained for the full cycler. The figure shows both the initial guess obtained in Section IV and the time-optimal trajectory. ...This paper investigates solar sail Earth-Mars cyclers, in particular cyclers between libration point orbits at the Earth-Moon L2 point and the Sun-Mars L1 point. In order to facilitate cyclers in as few Earth-Mars synodic periods as possible, the overall objective is to minimize the time of flight. These time-optimal cyclers are obtained by using a direct pseudospectral method and exploiting techniques from dynamical systems theory to obtain an initial guess. In particular, heteroclinic connections between the unstable and stable manifolds of the target libration point orbits at the Earth-Moon L2 point and the Sun-Mars L1 point are sought for. While such connections do not exist in the ballistic case, they can be achieved by complementing the dynamics with a solar sail and assuming a constant attitude of the sail with respect to the direction of sunlight. These trajectories are sub-optimal due to the assumed constant sail attitude as well as minor discontinuities in position and velocity at the linkage of the manifolds, which are overcome by transferring the initial guess to the direct pseudospectral optimal control solver. For near- to mid-term sails, results show time-optimal round-trip trajectories that span three synodic Earth-Mars periods, with a few months to one year stay times at the libration point orbits, depending on the time of departure within a five-month window. Through the propellant-less nature of solar sailing, these Earth-Mars cyclers can, in theory, be maintained indefinitely.
- ... The mass of primary and secondary primary bodies are denoted by m 1 and m 2 , respectively, but the mass of spacecraft is supposed to be negligible. In the rotating barycentric reference system and with the units of length, mass, and time defined in the work of Szebehely [25], the primary body is placed at −μ; 0; 0 and the secondary primary body is placed at 1 − μ; 0; 0, with a mass ratio μ m 2 ∕ m 1 m 2 . Using X to denote the Cartesian state of the spacecraft in the dimensionless rotating coordinates, the equation of motion of the spacecraft can be expressed as [25] ...... In the rotating barycentric reference system and with the units of length, mass, and time defined in the work of Szebehely [25], the primary body is placed at −μ; 0; 0 and the secondary primary body is placed at 1 − μ; 0; 0, with a mass ratio μ m 2 ∕ m 1 m 2 . Using X to denote the Cartesian state of the spacecraft in the dimensionless rotating coordinates, the equation of motion of the spacecraft can be expressed as [25] ...In this paper, the trajectory correction maneuver (TCM) problem of several lunar flyby transfers is investigated in the ephemeris model. The error analysis indicates that the state error of a transfer orbit will be enlarged by lunar flyby; hence, a TCM mustbeexecuted before lunar flyby. ATCM approach basedonthe shooting methodisproposed in the ephemeris model. This method isthen applied to design the TCMof two kindsoflunar flyby transfers: transfers between the sun-Earth/moon libration point orbits, and transfers from the Earth to the sun-Earth/moon libration point orbits. Some regions with abnormally large fuel costs are found and explained by the numerical method. Furthermore, several TCM strategies of lunar flyby transfers under practical constraints are presented and discussed. Finally, the effect of the navigation errors on the TCM problem is investigated.
- ... The scope of this Chapter is to review the restricted three-body problem theory, in order to provide the theoretical background necessary for the study of the relative motion in this type of scenarios. The results presented in this Chapter are based on the classical references [61][62][63]. ...... Szebehely provides a series expansion of the sought solutions [61]: de- noting with γ 1 and γ 2 the distances from L 1 and L 2 to the smaller primary, these can be expressed as ...The objective of the present Thesis is a detailed study of the relative motion dynamics and control in a two-body and a three-body gravity fields. For the former scenario, a general set of equations for the inclusion of arbitrary orbital perturbations is derived. The equations are then used for the design of a nonlinear H-infinity controller based on the state-dependent Riccati equation technique. A closed-form solution for the H-infinity control problem for the relative motion control on elliptic orbits is also presented, based on a linearized time-varying set of equations. Relative motion in the three-body scenario is also studied. In particular, a nonlinear set of equations for relative motion description in the local-vertical local-horizon frame is derived. Starting from this set, simplified equations are proposed and their performance compared. The computation of rendezvous maneuvers adopting both impulses or continuous thrust is then presented, in order to establish potential feasible trajectories.
- ... The system is parameterized by the mass ratio, µ = M 2 /(M 1 +M 2 ), where M 1 and M 2 are the masses of the primaries and masses, P 1 and P 2 , proceed on circular orbits about their mutual barycenter, B. The behavior of a third, relatively massless particle is described via the rotating coordinate frame, (ˆ x, ˆ y, ˆ z) P 1 and P 2 is unity, the mean motion of the two pri- maries is unity, and the masses of each body range from zero to one. 36 The equations of motion governing the CR3BP are derived via a Hamiltonian energy approach. Let the kinetic (T ) and potential (V ) energies corresponding to the CR3BP system be defined by ...... Global invariant manifolds are con- structed by transitioning the linear results to the non- linear model, where they are leveraged for trajec- tory design. 36 The CR3BP admits five equilibrium solutions, the well known Lagrange points. Three collinear points, i.e., L 1 , L 2 , and L 3 , are located on the rotating x-axis and are characterized by a saddle and a center subspace. ...A primary challenge of low-thrust mission design is the development of an initial guess for the state and control history of a trajectory. To address this challenge, one technique assembles dynamical structures, such as periodic orbits and their associated manifolds, into discontinuous chains that are corrected to locally optimal transfer solutions via direct transcription. In this investigation, dynamical structures that leverage a low-thrust force in a multi-body regime are incorporated into the orbit chain to expand the options available for construction of an initial guess and to guide the direct transcription algorithm toward different categories of locally optimal solutions. The properties and structures of the low-thrust model that facilitate orbit chain construction are demonstrated in representative transfer scenarios. Direct transcription is applied to converge upon locally optimal transfer trajectories in both simplified and ephemeris models. Results indicate that low-thrust dynamical structures offer a promising catalog of options in an orbit chain approach for designing optimal low-thrust trajectories.
- ... The restricted problem of three bodies has been widely studied, so we will only go through a very brief introduction here. For further and more thorough inspection of the equations refer to works such as [5] or [12]. ...In this paper we show the existence of syzygies for all periodic orbits inside the bounded Hill's region of the planer circular restricted three-body problem with energy below the second critical value. The proof will follow some ideas of Birkhoff to compute the roots of partial derivatives of the effective potential. Birkhoff's methods are extended to higher energies and a new base case is created and shown to fulfil the requirements. An other step from Birkhoff is scrutinized to continue the statement to all mass ratios. The final step is achieved by integrating over periodic orbits. Applying the same methods to Hill's lunar problem delivers similar results in that setting as well.
- ... Fig.(2.1) illustrates the po- sition of the third body m 3 referring to the center of mass of m 1 and m 2 , and the reference plane (x, y, z). The equations of motion for third body in syn- odic barycentric coordinate are given by (Szebehely, 1967): ...
- ... A trajectory design using a three-body dynamics plays a key role in realizing low energy trans- fers. 1,2 It enables to insert/leave into/from a periodic orbit around equilibrium points with very small fuel consumption using an invariant manifold. [3][4][5][6] For the invariant manifold, dynamical structures related to an optimal trajectory were studied. ...A structure of the optimal trajectory for minimizing fuel consumption in an unstable dynamical environment such as the three-body problem is not well studied. Recently, it has been found that a sparse solution structure appears in the optimal control of a dynamical system. The concept of sparsity explains the property that the minimum fuel trajectory corresponds to the trajectory which minimizes the total thrusting time. In this paper, we propose a numerical method to obtain the minimum fuel sparse optimal trajectory in the unstable dynamical system. As an example, proposed methods are applied to the transfer in the Sun-Earth system.
- ... Um caso especial desse problema ? o Problema Res- trito de Tr?s Corpos (PRCT) [6,7]. Trata-se de uma simplifica??o do problema geral de tr?s corpos e que ? ...Resumo Após uma revisão na literatura vimos que o Problema Restrito de Três Corpos (PRTC) é um tema com algumas publicações voltadas, principalmente, para a resolução do problema matemático, com muito poucas aplicações em situações reais e contextualizadas. Além disso, ao analisarmos alguns livros textos de mecânica clássica utilizados nos cursos de física, observamos que o PRTC não é desenvolvido e explorado didaticamente. Apoiados nessas análises o nosso objetivo neste trabalho é apresentar, desenvolver e aplicar o PRTC para simular as órbitas de dois astros do sistema solar: a do asteroide troiano da Terra, 2010 TK7, e a libração de Plutão devido a Netuno. Para resolver as equações diferencias do PRTC utilizamos o método de Cauchy em um código escrito na linguagem de programação Python. Nos dois casos estudados conseguimos verificar as trajetórias, valores e comportamentos orbitais dos astros. Por isso, entendemos que o PRTC pode ser aplicado para simular o movimento de diversos astros do sistema solar e, consequentemente, ser utilizado como um recurso pedagógico para contextualizar conceitos de mecânica celeste e dinâmica orbital. Uma vez que ele se relaciona com o estudo das equações diferencias, com a gravitação universal e com os métodos numéricos.
- ... As it is well known, one of the most famous models in non- linear physics was the perturbed harmonic oscillator because it contains nonlinear behavior that allows to test the different mod- ern theories on dynamic systems, as well as its theoretical and experimental applicability in several fields such as particle and plasmas physics [13,14], dynamic astronomy [15][16][17] and atomic physics [18,19]. ...Integrability in the Painlevé sense of the trapped ionic system in the quadrupole field with superpositions of rotationally symmetric hexapole and octopole fields is studied. Five integrable cases of the system are reported. First Integrals of the planar motion are founded. Confirming three-dimensional integrability of the equations of motion, the third explicit integrals of motion are constructed directly for each case. We carried out a numerical study to observe the regularity and chaotic regions via the Poincaré surface of sections, and corroborate the analytical results.
- ... Szebehely [1]. Many perturbing forces, like oblateness, radiation forces of the primaries, Coriolis and centrifugal forces etc., have been included in the study of the R3BP. ...
- ... Mars-Phobos DROs are periodic orbits obtained from the formulation of the Circular Restricted Three-Body Problem (CR3BP; general geometry shown in Figure 1), i.e. a dynamical system in which the gravity of two primary masses, m 1 and m 2 , act on each other and on a much smaller mass, m 3 , e.g. a spacecraft, which has no influence on the first two masses. 9 Furthermore, m 1 and m 2 orbit each other in circular orbits and the distance between them is R and rate of rotation is ˙ θ. Note that in the CR3BP, both R and ˙ θ are constant. ...This paper discusses a semi-analytical approach to approximate the relative motion of spacecraft around Distant Retrograde Orbits (DROs) in the vicinity of Phobos. Numerical analysis reveals that Mars-Phobos DROs can be represented in a moderately accurate way using second-order Fourier series representations. Assuming that the relative distance between spacecraft is small, the equations of motions can be simplified and semi-analytically solved to obtain a representation of the relative motion in the vicinity of Phobos. These equations exhibit a secular behavior, accurately described by the Hill-Clohessy-Wiltshire equations, and a short-term cyclic behavior due to the gravity of Phobos.
- ... The motion of the planets around the Sun and the stability of these motions was addressed by many mathematicians and physicists [1,2,3,4,5] and is still the subject of many current research papers [6,7,8,9,10,11]. At the core of this problem is the treatment of n-body interactions where n is large [1,10]. ...Fractional calculus made important contributions in many areas of applied mathematics in the last decade. A novel aspect of these applications is the inclusion of nonlocal effects on the evolution of the dynamical system under consideration. Using fractional calculus, these effects can be incorporated naturally in the differential equations governing the system evolution. In this paper, we illustrate the possible application of fractional calculus to some problem in celestial mechanics. To this end, we consider the computation of satellites orbits around the Earth. Usually, these computations ignore nonlocal effects (e.g. the impact of the Moon on the orbit) or at best include these as a perturbation. The present paper explores the possible use of fractional calculus to account for these nonlocal influences as an alternative paradigm to perturbations. In particular, we attempt to account for the effect of quadratic drag on satellite orbits using a fractional calculus model.
- ... This problem has five libration points: three of them are called the collinear points 1 [2]. Several studies have been carried out on the collinear libration points by considering the oblateness of one or two primaries for the circular restricted three-body problem. ...In the present work, the collinear equilibrium points of the restricted three-body problem are studied under the effect of oblateness of the bigger primary using an analytical and numerical approach. The periodic orbits around these points are investigated for the Earth-Moon system. The Lissajous orbits and the phase spaces are obtained under the effect of oblateness.
- ... where κ is the non-oscillatory poles nulling factor, and ω xy and ω z are the in-plane and out-of-plane frequencies, respectively. 22 The following values were chosen in order to simulate a 6-months orbit with mean radius equal to 60 000 km: ...The use of the state-dependent Riccati equation (SDRE) technique for formation flying control in the circular restricted three-body problem is investigated in this paper. First, the relative dynamics of a leader-follower formation is described by computing the difference between the equations of motion associated to the two spacecraft. Then, a pseudo-linear form of the relative motion equations is identified in order to implement the SDRE control technique. The effectiveness of the controller is proved by the high accuracy and the limited control usage achieved during the numerical simulations, set up considering the New Worlds Observer mission scenario.
- ... To do so, the coordinate system needs to rotate with the primaries, but also to 'pulsate', to account for the movement towards and away from the center of mass as the primaries travel along the ellipses. We refer the reader to [73] for details. ...We consider a class of autonomous Hamiltonian systems subject to small, time-periodic perturbations. When the perturbation parameter is set to zero, the energy of the system is preserved. This is no longer the case when the perturbation parameter is non-zero. We describe a topological method to show that for every suitably small, non-zero perturbation parameter, there exist diffusing orbits along which the energy changes by an amount independent of the perturbation, as well as orbits along which the energy makes chaotic jumps. The method yields quantitative estimates: an explicit range of the perturbation parameter for which these phenomena occur, the speed at which the energy changes along diffusing orbits, and the Hausdorff dimension of the set of initial conditions that exhibit chaotic behavior. In addition, the distributions of energies along orbits starting from some sets of initial conditions converge to a Brownian motion with drift as the perturbation parameter tends to zero. Moreover, we can obtain any desired values of the drift and of the variance for the limiting Brownian motion, for appropriate sets of initial conditions. Our results address some conjectures made by Arnold and Chirikov. A key feature of our topological method is that it can be implemented in computer assisted proofs. As an application, we show the existence of Arnold diffusion, and provide quantitative estimates, in a concrete model of the planar elliptic restricted three-body problem describing the motion of an infinitesimal body relative to the Neptune-Triton system.
- ... Halo orbits are periodic orbits obtained from the formulation of the Circular Restricted Three- Body Problem (CR3BP; general geometry shown in Figure 1), i.e. a dynamical system in which the gravity of two primary masses, m 1 and m 2 , act on each other and on a much smaller mass, m 3 , e.g. a spacecraft, which has no influence on the first two masses. 3,4 Furthermore, m 1 and m 2 orbit each other at distance R and rate of rotation ˙ θ. Note that in the CR3BP, both R and ˙ θ are constant. ...In this paper a simple and efficient way of computing impulsive maneuver transfers from a user-defined Low Earth Orbit (LEO) to a desired lunar halo orbit around the Earth-Moon Lagrange point 2 (EML2) utilizing a heuristic optimization method is presented. Fireworks, a hybrid heuristic optimization method, is utilized in order to obtain solutions with acceptable fidelity. The dynamical framework utilized is the Circular Restricted Three-Body Problem (CR3BP). Sample LEO to lunar halo trajectories along with their required delta-v costs and Time-of-Flight (TOF) are provided and compared to known numerical techniques to assert the validity of the proposed method.
- ... In addition, let ¯ g and ¯ λ be optimal solutions of [23], ...The problem of fixed-time fuel-optimal trajectories with high-thrust propulsion in the vicinity of a Lagrange point is tackled via the linear version of the primer vector theory. More precisely, the proximity to a Lagrange point i.e. any equilibrium point-stable or not-in the circular restricted three-body problem allows for a linearization of the dynamics. Furthermore, it is assumed that the spacecraft has ungimbaled thrusters, leading to a formulation of the cost function with the 1-norm for space coordinates, even though a generalization exists for steerable thrust and the 2-norm. In this context, the primer vector theory gives necessary and sufficient optimality conditions for admissible solutions to two-value boundary problems. Similarly to the case of rendezvous in the restricted two-body problem, the in-plane and out-of-plane trajectories being uncoupled, they can be treated independently. As a matter of fact, the out-of-plane dynamics is simple enough for the optimal control problem to be solved analytically via this indirect approach. As for the in-plane dynamics, the primer vector solution of the so-called primal problem is derived by solving a hierarchy of linear programs, as proposed recently for the aforementioned rendezvous. The optimal thrusting strategy is then numerically obtained from the necessary and sufficient conditions. Finally, in-plane and out-of-plane control laws are combined to form the complete 3-D fuel-optimal solution. Results are compared to the direct approach that consists in working on a discrete set of times in order to perform optimization in finite dimension. Examples are provided near various Lagrange points in the Sun-Earth and Earth-Moon systems, hinting at the extensive span of possible applications of this technique in station-keeping as well as mission analysis, for instance when connecting manifolds to achieve escape or capture.
- ... The CR3BP is typically represented in the synodic reference frame; all the physical quantities are normalized in such a way that both the sum of the primaries' masses and the distance that separates them is equal to 1 [30]. The normalized position and velocity in this model obey the following equations of motion: ...The design of a space trajectory is strongly linked to the gravitational and non-gravitational environment and the dynamical frameworks required to model it. These dynamical models may range from low to high fidelity, with corresponding computational costs. This paper proposes a multifidelity approach for the computation of nearly resonant trajectories with the Earth. This framework is used to compute trajectories for the capture of near-Earth asteroids into libration point orbits of the Sun–Earth system. The transfer is first computed in a suitable low-fidelity model, the Keplerian map, and a multifidelity approach is subsequently used to refine the solution from an impulsive approximation into a low-thrust transfer in the circular restricted three-body problem. The entire trajectory follows a nearly resonant motion with the Earth, lasting less than two synodic periods; starting when the retrieval spacecraft attaches itself to the asteroid, they will encounter the Earth twice, being captured into the target orbit at the end of the second encounter. A velocity change maneuver is carried out at the beginning of the motion, so that the first encounter with the Earth provides a gravitational perturbation resulting on a reduction of overall propellant costs of the transfer. The developed framework is very flexible in terms of the desired accuracy and allows for the low computational cost exploration of a vast number of possible trajectories. The obtained low-thrust transfers yield, for six asteroids, a much higher retrievable mass in comparison with direct capture trajectories, which do not undertake Earth-resonant encounters.
- ... A mathematical model that follows the assump- tions of the elliptic planar restricted three body problem (Szebehely 1967) is used to describe the mo- tion of a spacecraft around a system formed by two other bodies. One of these bodies has the largest mass of the system, and it is called the primary body (Mars), while the other one has a much smaller mass and is called the secondary body (Phobos). ...The goal of the present paper is to search and study mid-range planar orbits for a spacecraft traveling near Phobos. The first step is to make a numerical search and classification of natural orbits based in the concept of “Quasi Satellite Orbits” (QSO). The effects of the eccentricity of Phobos and the irregular shape of the bodies involved (Mars and Phobos) are studied, identifying the importance of these terms. This study is made using two different initial locations for Phobos, the periapsis and apoapsis. The results show the existence of several solutions, mapping the minimum, maximum and averaged Phobos-spacecraft distances. � Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México.
- ... Such support may come in the form of landing missions [1,2], lunar far- side communication capabilities [3,4], or as a gateway to more distant interplanetary destinations [1,5,6]. The natural motion around the libration points has been studied in great detail [7][8][9] and several families of (quasi-)periodic orbits around the libration points have been identified, e.g., Lissajous [10], Lyapunov [11], and halo [12] orbits, with more families in, for example, Kazantzis [13,14]. Though of immense importance, the fact that the spacecraft dynamics in these works are fully governed by gravitational accelerations only leaves little flexibility. ...This paper explores the existence of homo- and heteroclinic connections between solar-sail periodic orbits in the planar Earth-Moon circular restricted three-body problem. The existence of such connections has been demonstrated to great extent for the planar and spatial classical (no-solar sail) three-body problem, but remains unexplored for the inclusion of a solar-sail induced acceleration. Similar to the search for homo- and heteroclinic connections in the classical case, this paper uses the tools and techniques of dynamical systems theory, in particular trajectories along the unstable and stable manifolds, to generate these connections. However, due to the time dependency introduced by the solar-sail induced acceleration, common methods and techniques to find homo- and heteroclinic connections (e.g., using the Jacobi constant and applying spatial Poincaré sections) do not necessarily apply. The aim of this paper is therefore to gain an understanding of the extent to which these tools do apply, define new tools (e.g., solar-sail assisted manifolds, temporal Poincaré sections, and a genetic algorithm approach), and ultimately find the sought for homo- and heteroclinic connections. As a starting point of such an investigation, this paper focuses on the planar case, in particular on the search for homo- and heteroclinic connections between three specific solar-sail Lyapunov orbits (two at the L1 point and one at the L2 point) that all exist for the same near-term solar-sail technology. The results of the paper show that, by using a simple solar-sail steering law, where a piece-wise constant sail attitude is applied in the unstable and stable solar-sail manifold trajectories, homo- and heteroclinic connections exist for these three solar-sail Lyapunov orbits. The remaining errors on the position and velocity at linkage of the stable and unstable manifold trajectories are < 10 km and < 1 m/s. Future studies can apply the tools and techniques developed in this paper to extend the search for homo- and heteroclinic connections to other solar-sail Lyapunov orbits in the Earth-Moon system (e.g., for different solar-sail technology), to other planar solar-sail periodic orbits, and ultimately also to the spatial, three-dimensional case.
- ... The whole simulations are performed in a syn- odic reference frame, considering the effect of both asteroids on the CubeSat's trajectory. The distance between the two bodies is assumed to be constant, therefore the dynamics environment is the Circular Restricted Three Body Problem (CRTBP) [5]. Al- though the irregular shape of the bodies would cause small perturbations on the trajectories, the charac- teristic times of the flyby are so small that a point mass model represents a very good approximation. ...Although just few tens of binary asteroid systems are known, still they represent an intriguing natural facility for both planetary and universe science further understanding and technology in orbit demonstration. In particular, one of those, the Didymos system, recently captured the space community interest as a perfect target to test capabilities in detecting natural objects, for planetary protection. Indeed, JHU/APL, supported by NASA, are designing the Double Asteroid Redirection Test (DART) mission to impact on the Didymos secondary moon (Didymos B), and assess the kinetic impactor strategy performance to deflect a 150 m wide small asteroid. The impact ejecta, being DART a single spacecraft mission, will be monitored only remotely from Earth. However, to possibly be in close view of the impact point just before and after the kinetic event occurrence would offer the chance to collect unique scientific data: potential fragmentation of Didymos B could be registered, and plume material in situ analyzed. A simple plume evolution imaging may even offer fundamental information on the natural bodies composition and the deflection effectiveness. Assuming the possibility for the main spacecraft to host a small piggyback nanosat, the paper assesses the science opportunities offered by releasing the nanosat at the Didymos system arrival, to witness the impact and post-impact events in the Didymos B proximity. The time-to-impact nanosat release, the release relative velocity direction and magnitude are assumed as degrees of freedom to generate families of trajectories to maximize the post-impact environment monitoring, under the multi-body gravitational field of the binary system. The effectiveness of a low authority on board propulsion unit is also considered to widen the trade space for the nanosat trajectories in the binary proximity which maximize the time of residence in the impact region vicinity. Analyses showed that the nanosat trajectory can be tuned so that the impact expected fragments can be imaged from different perspectives, making the piggyback nanosat a very interesting added value to the kinetic impactor mission. The paper synthesizes the different opportunities that the proposed piggyback cubesat offers if the limited engineering and operational degrees of freedom merged with the peculiar gravity field are carefully exploited.
- ... The fact that tens of orbits is too short a time span to re- veal the underlying stationary behavior the accretion flow can be illustrated by repeating the map-stacking analysis for the e b = 0.6 case (see Figure 10; cf. Figure 6). The stacking of these images is trickier than in the circular case: in order for the two "stars" to line-up at all snapshots, we need to intro- duce a rotating-pulsating coordinate system, a transformation that is well known in the study of the elliptical restricted three- body problem (ER3BP; e.g., Nechvíle 1926;Kopal & Lyttleton 1963;Szebehely & Giacaglia 1964;Szebehely 1967;Musielak & Quarles 2014). The scaled, rotated coordinates ξ and η are related to the barycentric coordinates x and y by: ...We carry out 2D viscous hydrodynamical simulations of circumbinary accretion using the AREPO code. We self-consistently compute the accretion flow over a wide range of spatial scales, from the circumbinary disk far from the central binary, through accretion streamers, to the disks around individual binary components, resolving the flow down to 2% of the binary separation. We focus on equal mass binaries with arbitrary eccentricities. We evolve the flow over long (viscous) timescales until a quasi-steady is reached, in which the mass supply rate at large distances $\dot M_0$ (assumed constant) equals the time averaged mass transfer rate across the disk and the total mass accretion rate onto the binary components. This quasi-steady state allows us to compute the secular angular momentum transfer rate onto the binary, $\langle \dot J_b\rangle$, and the resulting orbital evolution. Through direct computation of the gravitational and accretion torques on the binary, we find that $\langle \dot J_b\rangle$ is consistently positive (i.e., the binary gains angular momentum), with $l_0\equiv \langle \dot J_b\rangle/\dot M_0$ in range of $(0.4-0.8)a_b^2\Omega_b$, depending on the binary eccentricity (where $a_b,~\Omega_b$ are the binary semi-major axis and angular frequency); we also find that this $\langle \dot J_b\rangle$ is equal to the net angular momentum current across the circumbinary disk, indicating that global angular momentum balance is achieved in our simulations. We compute the time-averaged rate of change of the binary orbital energy for eccentric binaries, and thus obtain the secular rates $\langle \dot a_b\rangle$ and $\langle \dot e_b\rangle$. In all cases, $\langle \dot a_b\rangle$ is positive, i.e., the binary expands while accreting. We discuss the implications of our results for the merger of supermassive binary black holes and for the formation of close stellar binaries.
- ... To investigate the motion of a massless particle near a binary system, it is convenient to utilize theoretical results and numerical methods that have been 15 used in the circular restricted three-body problem (CR3BP). The dynamical be- haviors in the CR3BP have been well studied in the past years [5,6,7,8,9,10]. ...Periodic orbits are important keys to understand the motion of a massless particle in the vicinity of a binary asteroid system. Due to the complex gravity generated by the irregular-shaped asteroids, it is difficult to generate periodic orbits with an analytical method, except with the linearized dynamics in a small region around the libration points. This paper has presented a numerical method to search the three-dimensional periodic orbits in a global space. The grid searching method is used to find nearly periodic orbits, and then the shooting method is used to get the accurate periodic orbits. The method is applied to the binary asteroid (66391) 1999 KW4 to solve the periodic orbits in the vicinity. The periodic orbits are then classified into five categories according to their shapes and locations. The search method can also be used to find periodic orbits in the vicinity of other binary systems.
- ... The dynamics of multi-body systems, and 3-body systems in particular, stem from the circular restricted 3-body problem, which describes the motion of an object around 2 other large bodies. Though this problem has been known for around 200 years, the formulation of the equations of motion were not well-described until 1967 [20], and one of the first missions to utilize them was the ISEE-3 which didn't launch until 1978 [6]. ...Today, Mars is one of the most interesting and important destinations for humankind and copious methods have been proposed to accomplish these future missions. One of the more fascinating methods is the Earth-Mars cycler trajectory which is a trajectory that accomplishes repeat access to Earth and Mars with little to no fuel-burning maneuvers. This would allow fast travel to and from Mars, as well as grant the possibility of multiple missions using the same main vehicle. Insertion from Earth-orbit onto the cycler trajectory has not been thoroughly ex- plored and the only existing method so far is a Hohmann-esque transfer via direct burn. The use of manifolds from gravitational equilibrium points has not been con- sidered for low energy transfer to the cycler trajectory. This work is primarily focused on closing this gap and analyzing the feasibility of this maneuver. To accomplish this, a study of the cycler trajectory – and the S1L1-B class specif- ically – was completed. The required gravity assist maneuvers at each planet was analyzed through V∞ matching and the entire trajectory was generated over the re- quired inertial period. This method allowed for the generation of 2 cycler trajectories of the inbound and outbound classes, which combine to allow for a reduction in the amount of time the astronauts spend in space. The Earth-Sun L2 point is analyzed as a potential hub for the maneuver and a halo orbit about this libration point is optimized for low energy transfer from and Earth parking orbit. The associated invariant manifold is then optimized for launch date and distance to the first trajectory on the cycler in order to burn from a trajectory on the manifold to the cycler trajectory. iv The comparisons of this work lie in the required ∆V to perform each maneuver compared to a direct burn onto the cycler trajectory. These values are compared and the practicality of this maneuver is drawn from these comparisons. It was found that the total required ∆V for the manifold method is larger than a direct burn from Earth orbit. However, this considers the trajectory from Earth to the halo orbit and if this is removed from consideration the ∆V is significantly reduced. It was shown that the feasibility of this method relies heavily on the starting position of the cycler vehicle. If the vehicle begins in Earth-orbit, a direct burn is preferred, however, if the vehicle began in a halo orbit (say it was assembled there) the manifold maneuver is largely preferable.
- 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.
- In this paper, computation of the halo orbit for the KS-regularized photogravitational circular restricted three-body problem is carried out. This work extends the idea of Srivastava et al. (2017) which only concentrated about the (i) regularization of the 3D-governing equations of motion, and (ii) validation of the modeling for small out-of-plane amplitude ( Az=110000 km) assuming the third-order analytical approximation as an initial guess with and without differential correction. This motivated us to compute the halo orbits for the large out-of-plane amplitudes and to study their stability analysis for the regularized motion. The stability indices are described as a function of out-of-plane amplitude, mass reduction factor, and oblateness coefficient. Three different Sun-planet systems: the Sun-Earth, Sun-Mars, and the Sun-Jupiter are chosen in this study. Stable halo orbits do not exist around the L1 point, however, around the L2 point stable halo orbits are found for the considered systems.
- In this paper we present how sample based analysis can complement classical methods for analysis of dynamical systems. We describe how sample based algorithms can be utilized to obtain better understanding of complex dynamical phenomena, especially in multistable dynamical systems that are difficult for analytical investigations. Relying on the simple, direct numerical integration algorithms we are able to detect all possible solutions including hidden and rare attractors; investigate the ranges of stability in multiple parameters space; analyse the influence of parameters mismatch or model imperfections; assess the risk of dangerous or unwanted behaviour and reveal the structure of multidimensional phase space. For each mentioned application we present methodology, example on paradigmatic non-linear dynamical system and discuss practical applications. The presented methods of analysis can be applied to solve numerous of scientific problems originating from different disciplines. Moreover, their robustness and efficiency will grow with the upcoming increase of computational power.
- A multiple grid search strategy is implemented to generate a broad database of axisymmetric three-body periodic orbits for planets and main planetary satellites in the Solar system. The periodic orbit search is performed over 24 pairs of bodies that are well approximated by the circular restricted three-body problem (CR3BP), resulting in approximately 3 million periodic solutions. The periodic orbit generation is implemented in a two-level grid search scheme. First, a global search is applied to each CR3BP system in order to capture the global structure of most existing families, followed by a local grid search, centered around a few fundamental families, where useful, highly sensitive periodic orbits emerge. A robust differential corrector is implemented with a full second-order trust region method in order to efficiently converge the highly sensitive solutions. The periodic orbit database includes solutions that (1) remain in the vicinity of the secondary only; (2) circulate the primary only via inner or outer resonances; and (3) connect both resonance types with orbits bound to the secondary, approximating heteroclinic connections that leads to natural escape/capture mechanisms. The periodic solutions are characterized and presented in detail using a descriptive nomenclature. Initial conditions, stability indices, and other dynamical parameters that allow for the solution characterization are computed and archived. The data and sample scripts are made available online.
- The fragmentation of interstellar molecular clouds has been investigated with great effort by many authors. In this paper, a simple model is given to describe the dynamics of two fragments moving in a special cylindrical potential. Using a modified version of the restricted three-body problem and the corresponding Jacobian integral, some constraints are given for the motion of the fragments.
- Spaceflight involving orbital transfers around irregularly shaped bodies or in the gravity field of several large bodies is fundamentally different from the flight in the gravity field of a single spherical body, which was covered in the previous chapters. The primary reason for this difference is that the spacecraft is no longer in a time-invariant gravity field of the two-body problem, but instead encounters a time-dependent field due to the relative motion of the multiple large bodies with respect to one another, or due to the changing position of the spacecraft relative to a rotating, non-spherical body.
- Using as reference test model the Planar Circular Restricted Three Body Problem , this paper explores its Lagrangian Coherent Structures, as well as its Hy-perbolic Lagrangian Coherent Structures. The purpose is to identify stable and unstable manifolds acting as separatrices between orbits with different qualitative behaviour and, therefore, relevant to the dynamics of the problem. Particular attention is given to the manifolds associated to the collinear libration points and to the practical stability regions around the triangular equilibrium points.
- In this work, periodic attitudes and bifurcations of periodic families are investigated for a rigid spacecraft moving on a stationary orbit around a uniformly rotating asteroid. Under the second degree and order gravity field of an asteroid, the dynamical model of attitude motion is formulated by truncating the integrals of inertia of the spacecraft at the second order. In this dynamical system, the equilibrium attitude has zero Euler angles. The linearised equations of attitude motion are utilised to study the stability of equilibrium attitude. It is found that there are three fundamental types of periodic attitude motions around a stable equilibrium attitude point. We explicitly present the linear solutions around a stable equilibrium attitude, which can be used to provide the initial guesses for computing the true periodic attitudes in the complete model. By means of a numerical approach, three fundamental families of periodic attitudes are studied, and their characteristic curves, distribution of eigenvalues, stability curves and stability distributions are determined. Interestingly, along the characteristic curves of the fundamental families, some critical points are found to exist, and these points correspond to tangent and period-doubling bifurcations. By means of a numerical approach, the bifurcated families of periodic attitudes are identified. The natural and bifurcated families constitute networks of periodic attitude families.
- This paper focuses on the trajectory design for LEO to Mars-Phobos DRO missions. Lunar DROs are also briefly explored as an alternative departure location. We present the methodology used to compute LEO to Mars-Phobos DRO trajectories and required launch C3, arrival v-inf, TOF, and total mission ΔV. Results show that using propellant-optimal LEO to Mars-Phobos DRO trajectories, such DROs could be used as a staging location between Mars and Phobos. Assuming refueling is available at the targeted DRO, LEO-LMO trajectories would have higher overall mission ΔV, but would have lower gear ratio thanks to the added “pit stop.”
- In applications of mechanics, including quantum mechanics, we often consider complex systems, where complete solutions of the underlying "fundamental" equations is both impractical and unnecessary to describe appropriate observations accurately. For example, practical chemistry, including even precision first-principles quantum chemistry, is never concerned with the behavior of the subnuclear quarks and gluons. Instead, we often focus on a few key variables, and construct a so-called effective theory for those. Such effective theories can become complicated and non-local, even for fairly simple systems. But in many circumstances, when there is a separation of scales, we can treat the reduced set of variables as a conventional dynamical system in its own right, governed by an energy conserving Lagrangian or Hamiltonian, in a useful approximation. The structure of that emergent description can display qualitatively new features, notably including reduced dimensionality, manifested through unconventional Poisson brackets. Here we discuss the physical meaning and consequences of such truncated dynamics. We propose physically realizable toy models of molecular rings, wherein time crystals emerge at the classical level. We propose that such behavior occurs in the effective theory of highly diamagnetic aromatic ring molecules, and could be widespread.
- This paper focuses on the trajectory design for missions destined to explore Mars and/or Phobos departing from Low Earth Orbit (LEO) and arriving into a Mars-Phobos Distant Retrograde Orbit (DRO). Lunar DROs are also briefly explored as an alternative departure location. A Mars-Phobos DRO is a relatively stable environment which would make both the surfaces of Mars and Phobos available for a reasonable propellant expenditure. This paper presents the methodology used to compute LEO to Mars-Phobos DRO trajectories and results regarding required C3 at launch, V-inf at arrival, Time-of-Flight (TOF), and total DeltaV for various Mars-Phobos DROs using full ephemeris planetary data. The results show that propellant-optimal trajectories from LEO to a specified Mars-Phobos DRO could be used as a staging location between Mars and Phobos. Assuming that refueling is available at the targeted DRO, LEO to Low Mars Orbits (LMO) trajectories would have higher total DeltaV due to the additional stop at the Mars-Phobos DRO. However, the aformentioned trajectories would have lower Initial Mass in LEO (IMLEO) and thus a lower gear ratio thanks to the added “pit stop” located at the given DRO. This results in a lower overall spacecraft dry mass that needs to be launched into space from Earth's surface.
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