Elliptic Restricted Three-body Problem Research Articles

Achieving solar sail orbital maintenance with adjustable ballast masses in the ERTBP

An innovative method for orbital maintenance in the Sun-Earth Elliptic Restricted Three-Body Problem (ERTBP) is introduced. The concept arose from observing that Natural Periodic (NP) orbits in the ERTBP exhibit deviations from their periodic behavior over time. These NP orbits are derived solely from accurately specified initial orbital conditions. Hence, a strategy for orbital maintenance becomes essential for these missions. This approach utilizes NP orbits as the designated reference trajectory to be tracked by the spacecraft. It assumes a solar sail equipped with two ballast masses and suggests a Fully Coupled Orbit-Attitude (FCOA) model. In this comprehensive model, both orbital and attitude states mutually influence each other, enabling orbital adjustments through attitude control. Therefore, the first step involves developing the governing equations for this FCOA model incorporating two ballast masses within the ERTBP. The strategy includes dividing the NP orbit period into τ equal time intervals. During each interval, adjusting the velocity of the ballast mass using a 7-mode electrical motor facilitates precise attitude control, establishing a framework for orbital maintenance. Maintaining constant velocity within each interval simplifies achieving the desired ballast mass positions. The adjustment of ballast mass velocity is proposed to be conducted using a hybrid meta-heuristic Invasive Weed Optimization/Particle Swarm Optimization (IWO/PSO). The analysis examines how increasing the mass of the ballast masses enhances tracking performance, underscoring the critical role of mass in optimizing tracking capabilities. This highlights the significance of meticulous structural design in achieving superior outcomes for orbital maintenance missions.

Location of the collinear equilibrium points in the elliptic restricted three-body problem with various perturbation effects

Abstract This study aims to examine the elliptic restricted three-body problem (ERTBP) by modifying the classical case and applying various perturbation sources to the three-body system. In this study, the locations of the Lagrange collinear equilibrium points of ERTBP were examined. We consider that the first primary body emits radiation and has an oblate shape. In contrast, the second primary body was considered to be elongated and approximated as a finite straight-segment. In addition, the perturbations from the disk-like structure around the three-body system were also included. The equations of motion of the infinitesimal body are presented in a dimensionless pulsating coordinate system. Three collinear equilibrium points were identified. The locations of the collinear equilibrium points were calculated numerically for several cases of perturbation values and also presented versus eccentricity over its range. We observed that the position of the collinear equilibrium points (L 1, L 2, and L 3) shifted when perturbing parameters were included, as opposed to where they were in the classical ERTBP.

A perturbative treatment of the Yarkovsky-driven drifts in the 2:1 mean motion resonance

Aims. Our aim is to gain a qualitative understanding as well as to perform a quantitative analysis of the interplay between the Yarkovsky effect and the Jovian 2:1 mean motion resonance under the planar elliptic restricted three-body problem. Methods. We adopted the semi-analytical perturbation method valid for arbitrary eccentricity to obtain the resonance structures inside the Jovian 2:1 resonance. We averaged the Yarkovsky force so it could be applied to the integrable approximations for the 2:1 resonance and the ν5 secular resonance. The rates of Yarkovsky-driven drifts in the action space were derived from the quasi-integrable approximations perturbed by the averaged Yarkovsky force. Pseudo-proper elements of test particles inside the 2:1 resonance were computed using N-body simulations incorporated with the Yarkovsky effect to verify the semi-analytical results. Results. In the planar elliptic restricted model, we identified two main types of systematic drifts in the action space: (Type I) for orbits not trapped in the ν5 resonance, the footprints are parallel to the resonance curve of the stable center of the 2:1 resonance; (Type II) for orbits trapped in the ν5 resonance, the footprints are parallel to the resonance curve of the stable center of the ν5 resonance. Using the semi-analytical perturbation method, a vector field in the action space corresponding to the two types of systematic drifts was derived. The Type I drift with small eccentricities and small libration amplitudes of 2:1 resonance can be modeled by a harmonic oscillator with a slowly varying parameter, for which an analytical treatment using the adiabatic invariant theory was carried out.

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In this paper, we present a way of combining the computation of invariant tori and their stable and unstable manifolds with the multiple shooting technique. We start by showing some of the results of Jorba (Nonlinearity 14(5):943–976, 2001) that should be modified in order to introduce the multiple shooting technique in these computations. After that, by a direct application in the planar elliptic restricted three-body problem (PERTBP), how to modify the equations and methods to compute the above-mentioned objects is introduced. In particular, the structure of the (systems of) equations and matrices involved in these computations is shown. An application of these computations can be found in Duarte and Jorba (Invariant manifolds of tori near L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}_1$$\\end{document} and L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}_2$$\\end{document} in the planar elliptic restricted three-body problem II. The Dynamics of Comet Oterma, Preprint 2023), where the dynamics of comet 39P/Oterma is modelled as a PERTBP.

Combined effect of Poynting-Robertson (P-R) drag, oblateness and radiation on the triangular points in the elliptic restricted three-body problem

This study investigates the motion of a test particle around triangular equilibrium points in the elliptic restricted three-body problem (ER3BP) under the influence of the two oblate and radiating primaries having Poynting-Robertson (P-R) drag. It is observed that the position of triangular points of the problem is affected by oblateness, radiation pressure, eccentricity, semi-major axis and Poynting-Robertson (P-R) drag. The stability of these points is demonstrated analytically by the Routh-Hurwitz criterion. It is seen that they are unstable under the combined effect of involved parameters. The effect of these parameters on the position of triangular points is examined numerically using the binary systems, 61 Cygni and Archird. The results obtained by these binary systems can be used to broaden the scope of interest in astronomy, astrophysics, space science and celestial mechanics in general.

Exploration and Maintenance of Homeomorphic Orbit Revs in the Elliptic Restricted Three-Body Problem

A novel station-keeping strategy leveraging periodic revolutions of homeomorphic orbits in the Elliptic Restricted Three-Body Problem within the pulsating frame is presented. A systemic approach founded on arc-length continuation is presented for the discovery, computation, and classification of periodic revolutions that morph from their traditional circular restricted three-body counterparts to build an a priori dataset. The dataset is comprehensive in covering all possible geometric architectures of the restricted problem. Shape similarity is quantified using Hausdorff distance and works as a filter for the station-keeping algorithm in relation to appropriate target conditions. Finally, an efficient scheme to quantify impulsive orbit maintenance maneuvers that minimize the total fuel cost is presented. The proposed approach is salient in its generic applicability across any elliptic three-body system and any periodic orbit family. Finally, average annual station-keeping costs using the described methodology are quantified for selected “orbits of interest” in the cis-lunar and the Sun–Earth systems. The robustness and efficacy of the approach instill confidence in its applicability for realistic mission design scenarios.

A Kustaanheimo–Stiefel regularization of the elliptic restricted three-body problem and the detection of close encounters with fast Lyapunov indicators

We present the Kustaanheimo–Stiefel (KS) regularization of the elliptic restricted three-body problem (ER3BP) at the secondary body P2, and discuss its use to study a category of transits through its Hill’s sphere (fast close encounters). Starting from the Hamiltonian representation of the problem using the synodic rotating–pulsating reference frame and the true anomaly of P2 as independent variable, we perform the regularization at the secondary body analogous to the circular case by applying the classical KS transformation and the iso-energetic reduction in an extended 10-dimensional phase-space; this translates into an efficient algorithm that can be readily implemented to numerically integrate the equations of motion. Using such a regularized Hamiltonian we recover a definition of fast close encounters in the ER3BP for small values of the mass parameter μ (while we do not require a smallness condition on the eccentricity of the primaries), and we show that for these encounters the solutions of the variational equations are characterized by an exponential growth during the fast transits through the Hill’s sphere. Thus, for small μ, we justify the effectiveness of the regularized fast Lyapunov indicators (RFLIs) to detect orbits with multiple fast close encounters. Finally, we provide numerical demonstrations and show the benefits of the regularization in terms of the computational cost.

Stable Orbiting Around Small Moons Using J2-Perturbed Elliptic Restricted Problem

Confirmed small-body missions Martian Moons eXploration (MMX) and Hera are set to explore Martian moons and the binary asteroid Didymos’s moon Dimorphos, respectively. Orbital dynamics around these small moons differ substantially from those around previously visited targets. Simplified models, such as the circular-restricted three-body problem, cannot yield accurate predictions for orbits and their stability in real-world operations. To be specific, the orbit of the small moon and its vicinity are significantly perturbed by the oblateness of the planet and their relative positions. Realistic control constraints and the unstable 3:1 resonance of retrograde orbits further complicate orbit maintenance around a small moon. Therefore, minimizing the dynamical perturbation on baseline orbits resulting from model mismatches is crucial. This paper introduces the J2-ER3BP+GH model dedicated to describing the orbital dynamics around the small moon. It incorporates the J2 perturbation of the planet on the elliptic-restricted three-body problem and can accommodate a nonspherical gravity field of the moon. Bounded orbits can still be identified without much effort in this sophisticated model. Baseline orbits around Phobos and Dimorphos from the J2-ER3BP+GH model become much easier to maintain, as verified in the high-fidelity dynamic and control environments.

Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem

Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem

Design of Flyby Trajectories with Powered Gravity and Aerogravity Assist Maneuvers

In this paper, we investigate flyby trajectories combining powered gravity assist (PGA) with aerogravity assist (AGA) in the planar elliptic restricted three-body problem (PERTBP). The patched flyby trajectory is divided into three portions: the PGA, AGA and ballistic portions, successively. In the PGA portion, continuous thrusts are conducted to change the speed and drive the altitude of a vehicle below the atmosphere edge. A simple flight-path angle guidance algorithm for three stages is used to design the orbit in the AGA portion. Taking the Sun–Mars PERTBP system as an example, flyby trajectories around Mars combining PGA with AGA are constructed and discussed in detail. In addition, numerical results show that the elliptical effect of the model should not be ignored, and it is necessary to investigate patched flyby orbits in the PERTBP.

Orbital precession in the restricted three-body problem: Exact representations

Analytical representations for the rate of apsidal precession in the planar elliptical restricted three-body problem are considered in the case when the orbit of the disturbing body is external to the orbit of the test particle. Analytical expressions are compared with numerical data obtained during numerous calculations of the precession rate. An analytical expression is obtained for the apsidal precession velocity in the form of a power series with a parameter equal to the ratio of the semi-major axes of the orbits of the test particle and the disturbing planet. It is shown that the analytical expressions for the velocity of the apsidal precession of the particle are reliable only at distances not close to the instability zone near the orbit of the disturbing planet. Near the Wisdom gap, the linear secular theory stops working. A correction empirical formula is proposed for calculating the apsidal precession rate at relatively high (less than 0.5) eccentricities of the particle and the disturbing planet. The proposed formulas are applied to the description of the precession of orbits in real exoplanetary systems.

Periodic Model Predictive Control for Tracking Halo Orbits in the Elliptic Restricted Three-Body Problem

Periodic Model Predictive Control for Tracking Halo Orbits in the Elliptic Restricted Three-Body Problem

Constraints to the semimajor axis of outer particles with nodal librations by general relativity effects

ABSTRACT We study nodal librations of outer particles in the framework of the elliptical restricted three-body problem including general relativity (GR) effects. From an analytical treatment based on secular interactions up to quadrupole level, we derive equations that define the nodal libration region of an outer test particle, which depends on the physical and orbital parameters of the bodies of the system. From this, we analyse how the GR constrains the semimajor axis of outer test particles that experience nodal librations under the effects of an inner planet around a single stellar component. Such an upper limit of the semimajor axis, which is called a2,lim, depends on the mass of the star ms, the mass m1, the semimajor axis a1, and the eccentricity e1 of the inner planet, and the eccentricity e2 of the outer test particle. On the one hand, our results show that the greater m1, a1, and e2 and the smaller ms, the greater the value of a2,lim. On the other hand, for fixed ms, m1, a1, and e2, a2,lim does not strongly depend on e1, except for large values of such an orbital parameter. We remark that N-body experiments of particular scenarios that include GR show results consistent with the analytical criteria derived in this research. Moreover, the study of hypothetical small body populations of real systems composed of a single star and an inner planetary-mass companion show that the GR effects can play a very important role in their global dynamics.

The Predictions of Noncollinear Equilibria Positions in ER3BP with Yukawa-Like Corrections

The existence and stability of noncollinear equilibrium points in the elliptic restricted three-body problem under the consideration of Yukawa correction to Newtonian potential are studied in this paper. The effects of various parameters (μ, ê, α, and λ) on the noncollinear equilibrium points are discussed briefly, and it is found that only ordinate of noncollinear equilibria E4,5 is affected by Yukawa correction while abscissa is affected by only mass parameter μ. The noncollinear equilibria was found linearly stable for a critical mass parameter μc. A critical point λ = ½ is also obtained for the critical mass parameter μc, and at this point, the critical mass parameter μc has maximum or minimum values according to α < 0 or α > 0, respectively.

Relegation-free closed-form perturbation theory and the domain of secular motions in the restricted three-body problem

We propose a closed-form (i.e., without expansion in the orbital eccentricities) scheme for computations in perturbation theory in the restricted three-body problem (R3BP) when the massless particle is in an orbit exterior to the one of the primary perturber. Starting with a multipole expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a sequence of normalizations in Delaunay variables by Lie series, leading to a secular Hamiltonian model without use of relegation. To this end, we introduce a book-keeping analogous to the one proposed in Cavallari and Efthymiopoulos (Celest Mech Dyn Astron 134(2):1–36, 2022) for test particle orbits interior to the one of the primary perturber, but here adapted, instead, to the case of exterior orbits. We give numerical examples of the performance of the method in both the planar circular and the spatial elliptic restricted three-body problem, for parameters pertinent to the Sun-Jupiter system. In particular, we demonstrate the method’s accuracy in terms of reproducibility of the orbital elements’ variations far from mean-motion resonances. As a basic outcome of the method, we show how, using as criterion the size of the series’ remainder, we reach to obtain an accurate semi-analytical estimate of the boundary (in the space of orbital elements) where the secular Hamiltonian model arrived at after eliminating the particle’s fast degree of freedom provides a valid approximation of the true dynamics.

Qualitative study of ballistic capture at Mars via Lagrangian descriptors

Lagrangian descriptors reveal the dynamical skeleton governing transport mechanisms of a generic flow. In doing so, they unveil geometrical structures in the phase space that separate regions with different qualitative behavior. This work investigates to what extent Lagrangian descriptors provide information about non-Keplerian motion in Mars proximity, which is modeled under the planar elliptic restricted three-body problem. We propose a novel technique to reveal ballistic capture orbits extracting separatrices of the phase space highlighted by Lagrangian descriptor scalar fields. The Roberts’ operator to approximate the gradient is used to detect the edges in the fields. Results demonstrate the chaos indicator ability to distinguish sets of initial conditions exhibiting different dynamics, including ballistic capture ones. Separatrices are validated against reference weak stability boundary derived on similar integration intervals. Compared to other techniques, Lagrangian descriptors provide dynamics insight bypassing the propagation of the variational equations.

Low-thrust transfer trajectory planning and tracking in the Earth–Moon elliptic restricted three-body problem

Low-thrust transfer trajectory planning and tracking in the Earth–Moon elliptic restricted three-body problem

Albedo effects in the ER3BP with an oblate primary, a triaxial secondary and a potential due to belt

We have examined the effects of Albedo in the Elliptic Restricted Three-Body Problem (ER3BP) with an oblate primary, a triaxial secondary, and potential due to belt for the Earth–Moon system. We have found that as the perturbed parameters increases, the possible boundary regions of the primary come closer to one other, allowing particles to travel from one region to the next freely and possibly merge the permissible regions. Our study has revealed that the formation of triangular libration points depends on the Albedo effects, semi-major axis, the Eccentricity of the orbits, triaxiality, and the potential due to the belt. As the parameters mentioned above increase, the triangular positions {L}_{4} and {L}_{5} move towards the center of origin in cases 1, 2, 3, and 4 and away from the center of the origin in cases 5, 6, and 7. Considering the range of a stable and unstable libration point for the problem under study given as 0<mu <{mu }_{c} for stable libration points and {mu }_{c}leqmu le frac{1}{2} for unstable libration points, our study has established that the triangular libration points are respectively stable and unstable for cases 1, 2, and 6 and cases 3, 4, 5, and 7. Our study has also revealed that each set of values has at least one characteristic complex root with a positive real part. Hence, the triangular libration points for the Earth–Moon system are unstable in the sense of Lyapunov. The Earth–Moon system's Poincare Surface of Section (PSS) has demonstrated that a slight change in the initial conditions, the semi-major axis, and the Eccentricity of the orbits have affected the system's behavior dramatically. Further, it is seen that a chaotic dynamical behavior of the system results into either regular or irregular orbits.

Non-integrability of the planar elliptic restricted three-body problem

We present the non-integrability proof for the planar elliptic restricted three-body problem. Two versions of this problem are considered: the classical one when only gravitational interactions are taken into account, and the photo-gravitational version where radiation pressure from the primaries is also included. Our result is valid for nonzero eccentricity and arbitrary mass ratio of the primaries. In the proof, we apply the differential Galois approach to study the integrability.

Continuation of some nearly circular symmetric periodic orbits in the elliptic restricted three-body problem

Continuation of some nearly circular symmetric periodic orbits in the elliptic restricted three-body problem