Bicircular Problem Research Articles

Transfers to Earth-Moon Triangular Libration Points by Sun-Perturbed Dynamics

This paper is devoted to the construction of transfers to Earth-Moon triangular libration points based on their Sun-perturbed dynamics in the Sun-Earth-Moon system. In this bi-circular problem (BCP), the solar gravity perturbation breaks the stability of the Earth-Moon triangular libration points with three remaining periodic orbits (two stable and one unstable) around them as their dynamical substitutions. As the stable region, the invariant tori full of quasi-periodic orbits around the two stable periodic orbits can be used as the destination of triangular libration point transfers for a low station-keeping cost. However, the instability of the unstable periodic orbit is so mild that its stable and unstable manifolds are folded around the invariant tori surrounding the other two stable orbits. Thus, the authors approximate the stable and unstable manifolds of the unstable orbit through the parameterization method, which emerge from the Sun-perturbed Earth-Moon triangular libration points vicinity. They act as a bridge between outer space and the stable region around triangular libration points. First, the transfers connecting the Earth’s vicinity are constructed using the manifolds. Subsequently, a new connection mechanism near L4 and L5 based on the N:M multi-loop transfers is developed by extending the conventional patching skill to time-dependent situations. The quickest one is achieved by a 1:1 L4-L5 heteroclinic-based connection with a TOF of less than 26.9 days at a fuel cost of 117.88 m/s. Note that such Sun-perturbed transfers cannot be produced by continuing any trajectory in the Earth-Moon circular restricted three-body problem but they can generate two-impulsive transfers in the ephemeris model by continuation.

Dynamics around the Earth–Moon triangular points in the Hill restricted 4-body problem

This paper investigates the motion of a small particle moving near the triangular points of the Earth–Moon system. The dynamics are modeled in the Hill restricted 4-body problem (HR4BP), which includes the effect of the Earth and Moon as in the circular restricted 3-body problem (CR3BP), as well as the direct and indirect effect of the Sun as a periodic time-dependent perturbation of the CR3BP. Due to the periodic perturbation, the triangular points of the CR3BP are no longer equilibrium solutions; rather, the triangular points are replaced by periodic orbits with the same period as the perturbation. Additionally, there is a 2:1 resonant periodic orbit that persists from the CR3BP into the HR4BP. In this work, we investigate the dynamics around these invariant objects by performing a center manifold reduction and computing families of 2-dimensional invariant tori and their linear normal behavior. We identify bifurcations and relationships between families. Mechanisms for transport between the Earth, L4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_4$$\\end{document}, and the Moon are discussed. Comparisons are made between the results presented here and in the bicircular problem (BCP).

Geometry of transit orbits in the periodically-perturbed restricted three-body problem

In the circular restricted three-body problem, low energy transit orbits are revealed by linearizing the governing differential equations about the collinear Lagrange points. This procedure fails when time-periodic perturbations are considered, such as perturbation due to the sun (i.e., the bicircular problem) or orbital eccentricity of the primaries. For the case of a time-periodic perturbation, the Lagrange point is replaced by a periodic orbit, equivalently viewed as a hyperbolic-elliptic fixed point of a symplectic map (the stroboscopic Poincaré map). Transit and non-transit orbits can be identified in the discrete map about the fixed point, in analogy with the geometric construction of Conley and McGehee about the index-1 saddle equilibrium point in the continuous dynamical system. Furthermore, though the continuous time system does not conserve the Hamiltonian energy (which is time-varying), the linearized map locally conserves a time-independent effective Hamiltonian function. We demonstrate that the phase space geometry of transit and non-transit orbits is preserved in going from the unperturbed to a periodically-perturbed situation, which carries over to the full nonlinear equations.

Transfers from the Earth to L_2\xa0Halo orbits in the Earth\u2013Moon bicircular problem

This paper deals with direct transfers from the Earth to Halo orbits related to the translunar point. The gravitational influence of the Sun as a fourth body is taken under consideration by means of the Bicircular Problem (BCP), which is a periodic time dependent perturbation of the Restricted Three Body Problem (RTBP) that includes the direct effect of the Sun on the spacecraft. In this model, the Halo family is quasi-periodic. Here we show how the effect of the Sun bends the stable manifolds of the quasi-periodic Halo orbits in a way that allows for direct transfers.

Lie series solution of the bicircular problem

The present work performs a semi–analytical solution for the orbit of an infinitesimal particle in the framework of the bicircular Sun–Earth–Moon system. In particular, Lie series technique is applied to find the solution of the equations of motion of bicircular Sun–Earth–Moon system with radiating bigger primary. To apply Lie-series technique, the second order system of ordinary differential equations has been reduced to the corresponding first order system. Then, a set of recurrence relations is obtained in the Lie series solutions of the bicircular model (BCM) and graphical representations of the orbit for short, intermediate and long time are shown. Moreover, we study the effect of the radiation parameter on the orbit of the massless body and demonstrate that this parameter as well as the initial conditions affect its size. Specifically, it is observed that the trajectory enlarges further when the values of the radiation parameter increase while additionally its size enlarges or compacts according to the selected set of initial conditions.

Using invariant manifolds to capture an asteroid near the L3 point of the Earth-Moon Bicircular model

This paper focuses on the capture of Near-Earth Asteroids (NEAs) in a neighbourhood of the L3 point of the Earth-Moon system. The dynamical model for the motion of the asteroid is the planar Earth-Moon-Sun Bicircular problem (BCP). It is known that the L3 point of the Restricted Three-Body Problem is replaced, in the BCP, by a periodic orbit of centre×saddle type, with a family of mildly hyperbolic tori that is born from the elliptic direction of this periodic orbit. It is remarkable that some pieces of the stable manifolds of these tori escape (backward in time) the Earth-Moon system and become nearly circular orbits around the Sun. In this work we compute this family of invariant tori and also high order approximations to their stable/unstable manifolds. We show how to use these manifolds to compute an impulsive transfer of a NEA to an invariant tori near L3. As an example, we study the capture of the asteroid 2006 RH120 in its approach of 2006. We show that there are several opportunities for this capture, with different costs. It is remarkable that one of them requires a Δv as low as 20 m/s.

Exploiting manifolds of [formula omitted] halo orbits for end-to-end Earth–Moon low-thrust trajectory design

In this work, end-to-end low-thrust transfers from a GTO orbit to a low-altitude lunar orbit by exploiting the manifolds of a chosen Earth–Moon L1 halo orbit has been studied. The practicality of piece-wise, minimum-time transfers that exploit halo orbit manifolds is demonstrated, which offers more flexibility to meet mission objectives. It is known that the structure of the manifolds varies substantially due to the presence of the Sun and its contribution has to be considered to obtain more realistic trajectories. To incorporate Sun’s perturbation, we study (1) manifolds’ behavior within a Bi-Circular Problem (BCP) dynamics and (2) Sun’s impact on the previously converged trajectories obtained using the standard Circular Restricted Three-Body Problem (CR3BP). Comparisons of the resulting trajectories using the CR3BP and BCP are presented.

Families of Halo-like invariant tori around $$L_2$$ in the Earth-Moon Bicircular Problem

The Bicircular Problem (BCP) is a periodic time dependent perturbation of the Earth-Moon Restricted Three-Body Problem that includes the direct gravitational effect of the Sun. In this paper we use the BCP to study the existence of Halo-like orbits around \(L_2\) in the Earth-Moon system taking into account the perturbation of the Sun. By means of computing families of 2D invariant tori, we show that there are at least two different families of Halo-like quasi-periodic orbits around \(L_2\).

Transport and invariant manifolds near L3 in the Earth-Moon Bicircular model

This paper focuses on the role of L3 to organise trajectories for a particle going from Earth to Moon and viceversa, and entering or leaving the Earth-Moon system. As a first model, we have considered the planar Bicircular problem to account for the gravitational effect of the Sun on the particle. The first step has been to compute a family of hyperbolic quasi-periodic orbits near L3. Then, the computation of their stable and unstable manifolds provides connections between Earth and Moon, and also generates trajectories that enter and leave the Earth-Moon system. Finally, by means of numerical simulations based on the JPL ephemeris we show that these connections can guide the journey of lunar ejecta towards the Earth.

The vicinity of the Earth–Moon $$L_1$$ point in the bicircular problem

The bicircular model is a periodic time-dependent perturbation of the Earth–Moon restricted three-body problem that includes the direct gravitational effect of the Sun on the infinitesimal particle. In this paper, we focus on the dynamics in the neighbourhood of the $$L_1$$ point of the Earth–Moon system. By means of a periodic time-dependent reduction to the centre manifold, we show the existence of two families of quasi-periodic Lyapunov orbits, one planar and one vertical. The planar Lyapunov family undergoes a (quasi-periodic) pitchfork bifurcation giving rise to two families of quasi-periodic halo orbits. Between them, there is a family of Lissajous quasi-periodic orbits, with three basic frequencies.

On the stabilizing effect of Solar Radiation Pressure in the Earth-Moon system

Solar sails change the natural dynamics of systems: Trajectories that are driven by gravitational forces can be displaced and changed because of the effect of Solar Radiation Pressure (SRP). Moreover, if the lightness number of the sail is large enough, the instability of certain orbits can be diminished and even removed. In this paper we modify two models for the motion of a probe in the Earth-Moon system that include the effect of Sun’s gravity to take also into account the effect of SRP. These models, the Bicircular Problem (BCP) and the Quasi-Bicircular Problem (QBCP), are periodic perturbations of the Earth-Moon Restricted Three Body Problem (RTBP). The models are modified to consider the effect of the SRP upon a solar sail. We provide examples of periodic orbits that are stabilized (or made less unstable) due to the effect of SRP.

Low-energy Lunar Trajectories with Lunar Flybystwo

The low-energy lunar trajectories with lunar flybys are investigated based on the Sun-Earth-Moon bicircular problem (BCP). The characteristics of the distribution of trajectories in the phase space are summarized. Using the invariant manifolds in the BCP system, the low-energy lunar trajectories with lunar flybys are sought. Then, take time as an augmented dimension in the phase space of a nonautonomous system, we present the state space map and reveal the distribution of these lunar trajectories in the phase space. Consequently, we find that the low-energy lunar trajectories exist as families, and that the every moment in the Sun-Earth-Moon synodic period can be the departure date. Finally, we analyse the velocity increment, transfer duration, and system energy for the different trajectory families, and obtain the velocity-impulse optimal family and the transfer-duration optimal family, respectively.

Two Periodic Models for the Earth-Moon System

This paper discusses two alternative models to the Restricted Three Body Problem (RTBP) for the study of a massless particle in the Earth-Moon system. These models are the Bicircular Problem (BCP) and the Quasi-Bicircular Problem (QBCP). While the RTBP is autonomous, the BCP and the QBCP are periodically time dependent due to the inclusion of the Sun’s gravitational potential. Each of the two alternative models is suitable for certain regions of the phase space. More concretely, we show that the BCP is more adequate to study the dynamics near the triangular points while the QBCP is more adequate for the dynamics near the collinear points.

Numerical investigation of transport mechanism in four-body problem using Lagrangian coherent structure

Transport mechanism is critical for understanding natural phenomena in the solar system and is beneficial to space mission design. In this study, transport mechanism in the bicircular four-body problem is numerically explored by using Lagrangian coherent structure (LCS), a tool widely used for identifying transport barriers in fluid flow. First, equations of motion of the bicircular problem are derived and five topology configurations of forbidden region are presented. Then, definition and computational method of LCS are introduced. Finally, properties of LCS which are useful for revealing transport mechanism in the four-body problem are numerically investigated.

Transfer orbits to the Earth–Moon triangular libration points

The particular positions and dynamics of the triangular libration points in the Earth–Moon system make them potential candidates for future space applications. Taking the leading L4 point as an example, this paper studies the transfer orbits to the vicinity of this equilibrium point. Two basic models are used: the circular restricted three-body problem (CRTBP) and the bi-circular problem (BCP). The order-three analytical solution of the motion around the triangular libration points in the CRTBP model is taken as the nominal orbit. Three different approaches are studied: direct transfer, transfer utilizing powered lunar gravity assist, and transfer utilizing the Sun’s gravity. Lastly, low energy transfer orbits are extensively studied in the BCP model via a numerical approach. Our studies show that the total delta-v cost is considerably reduced if the Moon’s gravity or the Sun’s gravity can be used, at the cost of a longer transfer time. The delta-v cost and the time of flight (TOF) in our work are approximately: 3.90–4.40km/s and 4–8days for the direct transfer; 3.41–3.48km/s and 18–52days for the transfer utilizing powered lunar gravity assist; 3.33–3.40km/s and 70–95days for the transfer utilizing the Sun’s gravity. For the low energy transfer orbits, the lower limit of the delta-v cost is around 3.10km/s.

Applications of lagrangian coherent structures to expression of invariant manifolds in astrodynamics

This paper investigates the relationship between invariant manifold and Lagrangian coherent structure (LCS) in dynamical systems. LCS is defined as the ridge of finite-time Lyapunov exponent (FTLE) field, and is proving to be excellent platform for studies of stable and unstable manifold in flows with arbitrary time dependence. In this study, the LCS tool is applied to autonomous systems, simple pendulum and planar circular restricted three-body problem (PCR3BP), and also non-autonomous ones, double-gyre flow and bicircular problem (BCP). A comparison between LCS and invariant manifold is presented.

Numerical and analytical study of the gravitational capture in the bicircular problem

Abstract The objective of the present paper is to study the ballistic gravitational capture in a dynamical model that has the presence of four bodies. In particular, the Earth–Moon–Sun–Spacecraft system is considered. This phenomenon is explained in terms of the integration of the perturbing forces with respect to time. Analytical equations are derived to estimate their magnitude and to show the best directions of approaching for the spacecraft. Using those equations, an analytical estimate of the fourth body effect valid for any system of primaries is derived. Numerical simulations are also performed to show the differences between the models with three and four bodies in the trajectories of the spacecraft.

On the vertical families of two-dimensional tori near the triangular points of the bicircular problem

This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system. The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the vertical (z and z) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov family of periodic orbits of the Lagrangian points that goes in the (z, z) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a very long time. It is remarkable that these regions seem to persist in the real system.

Study of the transfer from the Earth to a halo orbit around the equilibrium pointL 1

The purpose of this paper is to study a transfer strategy from the vicinity of the Earth to a halo orbit around the equilibrium pointL 1 of the Earth-Sun system. The study is done in the real solar system (we use the DE-118 JPL ephemeris in the simulations of motion) although some simplified models, such as the restricted three body problem (RTBP) and the bicircular problem, have been also used in order to clarify the geometrical aspects of the problem. The approach used in the paper makes use of the hyperbolic character of the halo orbits under consideration. The invariant stable manifold of these orbits enables the transfer to be achieved with, theoretically, only one manoeuvre: the one of insertion into the stable manifold. For the total Δv required, the figures obtained are similar to the ones given by the standard procedures of optimization.