Three-body Problem Research Articles

Quantitative kam Theory, with an Application to the Three-Body Problem

Based on quantitative “kam theory”, we state and prove two theorems about the continuation of maximal and whiskered quasi-periodic motions to slightly perturbed systems exhibiting proper degeneracy. Next, we apply such results to prove that, in the three-body problem, there is a small set in phase space where it is possible to detect both such families of tori. We also estimate the density of such motions in proper ambient spaces. Up to our knowledge, this is the first proof of co-existence of stable and whiskered tori in a physical system.

Analytic Understanding of the Resonant Nature of Kozai–Lidov Cycles with a Precessing Quadrupole Potential

The very long-term evolution of the hierarchical restricted three-body problem with a slightly aligned precessing quadrupole potential is studied analytically. This problem describes the evolution of a star and a planet that are perturbed either by a (circular and not too inclined) binary star system or by one other star and a second more distant star, as well as a perturbation by one distant star and the host galaxy or a compact-object binary system orbiting a massive black hole in nonspherical nuclear star clusters. Previous numerical experiments have shown that when the precession frequency is comparable to the Kozai–Lidov timescale, long-term evolution emerges that involves extremely high eccentricities with potential applications for a broad scope of astrophysical phenomena, including systems with merging black holes, neutron stars, or white dwarfs. By averaging the secular equations of motion over the Kozai–Lidov cycles (KLCs), we solve the problem analytically in the neighborhood of the KLC fixed point where the eccentricity vector is close to unity and aligned with the quadrupole axis and for a precession rate similar to the Kozai–Lidov timescale. In this regime the dynamics is dominated by a resonance between the perturbation frequency and the precession frequency of the eccentricity vector. While the quantitative evolution of the system is not reproduced by the solution far away from this fixed point, it sheds light on the qualitative behavior.

Low-Thrust Transfer Dynamics and Control Between Halo Orbits in the Earth–Moon System by Means of Invariant Manifold

The low-thrust transfer between halo orbits in the Earth-Moon system is studied in this paper. The optimization problem of low-thrust transfer is challenging in the circular restricted three-body problem (CRTBP). Making reasonable use of the three-body dynamics structure can reduce the thrust required for the transfer. The invariant manifolds associated with planar Lyapunov orbits are spliced at first to create homoclinic and heteroclinic connections that can be used as trial solutions for low-thrust transfer. After that, the transfer of spatial halo orbits is addressed using the continuation methods. The Pontryagin maximum principle is used to calculate the optimal control law for the transfer process. Trajectories with various numbers of rotations around the Moon are suitable for transfer between halo orbits with different orientations, depending on the category of the manifold splicing. Finally, a typical MPC controller is designed to track the transfer trajectory in the presence of perturbations. Numerical simulations show the efficiency and practicability of the optimal transfer trajectory design strategy by means of invariant manifold for future lunar exploration missions.

From Babylonian lunar observations to Floquet multipliers and Conley–Zehnder indices

In 1878, Hill found numerically, in his limiting case of the restricted three-body problem, the so-called Hill’s lunar problem, a planar direct periodic orbit with a period of one synodic month. By virtue of the spatial system’s invariance under a symplectic involution, whose fixed point set corresponds to the planar problem, we can assign to Hill’s orbit planar and spatial Floquet multipliers and planar and spatial Conley–Zehnder indices. We show that these have deep astronomical significance because, on the one hand, we relate the anomalistic month to the planar Floquet multipliers and the planar Conley–Zehnder index. On the other hand, we relate the draconitic month to the spatial Floquet multipliers and the spatial Conley–Zehnder index. Knowledge of this lunar month dates back to the Babylonians, who lived until around 500 BCE. In order to determine the indices, we analyze analytically the bifurcation procedure of the fundamental families of planar direct and retrograde periodic orbits (traditionally known as families g and f) from the rotating Kepler problem for very low energies.

Averaged Spatial Circular Restricted Three-Body Problem: Internal Case and New Results

Averaged Spatial Circular Restricted Three-Body Problem: Internal Case and New Results

Rendezvous and Proximity Operations in Cislunar Space Using Linearized Dynamics for Estimation

As interest in Moon exploration grows, and efforts to establish an orbiting outpost intensify, accurate modeling of spacecraft dynamics in cislunar space is becoming increasingly important. Contrary to satellites in Low Earth Orbit (LEO), where it takes around 5 ms to communicate back and forth with a ground station, it can take up to 2.4 s to communicate with satellites near the Moon. This delay in communication can make the difference between a successful docking and a catastrophic collision for a remotely controlled satellite. Moreover, due to the unstable nature of trajectories in cislunar space, it is necessary to design spacecraft that can autonomously make frequent maneuvers to stay on track with a reference orbit. The communication delay and unstable trajectories are exactly why autonomous navigation is critical for proximity operations and rendezvous and docking missions in cislunar space. Because spacecraft computational hardware is limited, reducing the computational complexity of navigational algorithms is both desirable and often necessary. By the introduction of a linear system approach to the deputy spacecraft motion, this research avoids the computational burden of integrating the deputy relative equations of motion. In this research, the relative CR3BP equations of motion are derived and linearized using a matrix exponential approximation. This research continues the development of the matrix exponential linearized relative circular restricted three-body problem (CR3BP) equations by applying the dynamics model to estimation and control applications. A simulation is performed to compare state estimation results obtained from using the linearized equations of motion utilizing a Kalman filter and for state estimation utilizing an unscented Kalman filter with the full nonlinear equations of motion. The linearized exponential model is shown to be sufficient for state estimation in the presence of noisy measurements for an example scenario. Additionally, a linear quadratic regulator (LQR) controller was added to optimally control a deputy spacecraft to rendezvous with a chief spacecraft in cislunar space. The contribution of this work is twofold: to provide a proof of concept that the matrix exponential solution for the linearized relative CR3BP equations can be used as the dynamics model for state estimation, as well as to simulate an optimal rendezvous maneuver in the presence of measurement noise.

Equilibrium points and Lyapunov families in the circular restricted three-body problem with an oblate primary and a synchronous rotating dipole secondary: Application to Luhman-16 binary system

We numerically study a version of the synchronous circular restricted three-body problem, where an infinitesimal mass body is moving under the Newtonian gravitational forces of two massive bodies. The primary body is an oblate spheroid while the secondary is an elongated asteroid of a combination of two equal masses forming a rotating dipole which is synchronous to the rotation of the primaries of the classic circular restricted three-body problem. In this paper, we systematically examine the existence, positions, and linear stability of the equilibrium points for various combinations of the model's parameters. We observe that the perturbing forces have significant effects on the positions and stability of the equilibrium points as well as the regions where the motion of the particle is allowed. The allowed regions of motion as determined by the zero-velocity surface and the corresponding isoenergetic curves as well as the positions of the equilibrium points are given. Finally, we numerically study the binary system Luhman-16 by computing the positions of the equilibria and their stability as well as the allowed regions of motion of the particle. The corresponding families of periodic orbits emanating from the collinear equilibrium points are computed along with their stability properties.

Formation of Cluster Systems in Chaotic Condensed Media

Purpose. The study of cluster formation in a system of chaotically moving and interacting particles taking into account the Efimov effect and the "golden" section.Methods. Methods of mathematical modeling, quantum mechanics, a model of solid spheres, and a cluster model were used.Results. Within the framework of the proposed work, it is noted that in a three-particle system of particles, it is possible to form their spatial configuration in the form of a "golden" triangle, and in the case of an excited state of two particles, the third particle is far enough away from the other two, it is this configuration that corresponds to the conditions for the occurrence of the Efimov effect in a three-particle system.Based on the mathematical formalism of the description of self-organization processes in the work, it is shown that in chaotic environments within the framework of the Efimov model, with the involvement of the "golden" section in the mutual arrangement of three interacting particles, it is possible to form disk-shaped clusters containing a "magic" number of particles. In the structure of these clusters, the formation of quantum-dimensional regions in the form of a torus is possible. The parameters of such areas are defined.Conclusion. The described model of the formation and decay of disk-shaped clusters, taking into account the Efimov effect and the "golden" section rule, allows us, without resorting to a complex solution of equations in the three-body problem, to obtain important relations following from strict theories. The proposed approach implies the possibility of self-organization of clusters and the formation of quantum-dimensional regions in their structure, for example, in the form of a torus with a potential well, capable of capturing charged particles and determining their energy spectrum, as well as explaining the appearance of spectral bands in the IR spectra of substances.The proposed approach may be of practical importance, for example, for predicting the IR spectra of liquids, the presence of quantum dots in liquids with a wide spectrum of excitation from UV to IR radiation.

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.

Formation flying orbits and GNC design in binary asteroid systems

Formation flying represents a promising opportunity for next space missions, due to its benefits in cost saving, redundancy and fault tolerance given by multiple small and cheap space segments exploitation, with comparable performance of a single, large monolithic spacecraft. That is even more relevant in asteroid exploration, as the harsh environment poses great risks to the probes survival. The low, irregular gravity field, the poor knowledge of shape and composition, and the possible presence of floating particles in the surroundings suggest adopting a low-risk strategy for the exploration of these bodies, delegating proximity operations to multiple nanosatellites, while keeping the main spacecraft at safe distance. As a downside, multiple cooperating spacecraft imply advanced capabilities in accurately reconstructing the relative positions and displacements and in fixing reconfiguration manoeuvres. Moreover, whenever relative distance is very close, agents’ guidance and control shall be autonomously computed by the spacecraft, as promptness of commands is not ensured relying on ground segment only. The partially unknown environment exacerbates this issue, demanding navigation and control schemes capable of withstanding potentially large uncertainties in the dynamics. If a binary system is considered, the multiple gravitational sources complicate even more the setup of the formation, requiring an accurate search and selection of operative orbits. This study will be thus defined by three different steps. First the search and development of suitable trajectories to host a reconfigurable satellite formation is carried out. Then the design of a Model Predictive on-board guidance and control law is performed to assess the capability of the spacecraft to execute successful reconfiguration transfers relying on a simplified dynamical model (namely the Circular Restricted Three-Body Problem). At the end the navigation error requirements are derived, in order to ensure the feasibility of the guidance and control scheme. Inspired by the Hera mission, this study explores the aforementioned aspects of the design of a formation for the exploration of the Didymos binary asteroid system, which comprises many complications but also challenges for many other future missions to fly in unexplored environments.

End-to-end trajectory concept for close exploration of Saturn’s Inner Large Moons

We present a trajectory concept for a small mission to the four inner large satellites of Saturn. Leveraging the high efficiency of electric propulsion, the concept enables orbit insertion around each of the moons, for arbitrarily long close observation periods. The mission starts with a EVVES interplanetary segment, where a combination of multiple gravity assists and deep space low thrust enables reduced relative arrival velocity at Saturn, followed by an unpowered capture via a sequence of resonant flybys with Titan. The transfers between moons use a low-thrust control law that connects unstable and stable branches of the invariant manifolds of planar Lyapunov orbits from the circular restricted three-body problem of each moon and Saturn. The exploration of the moons relies on homoclinic and heteroclinic connections of the Lyapunov orbits around the L1 and L2 equilibrium points. These science orbits can be extended for arbitrary lengths of time with negligible propellant usage. The strategy enables a comprehensive scientific exploration of the Inner Large Moons, located deeply inside the gravitational well of Saturn, which is unfeasible with conventional impulsive maneuvers due to excessive fuel consumption.

Classification of Trajectories in a Two-planet Exosystem Using the Generalized Three-body Problem

We deploy the theory of the generalized three-body problem to describe an exosystem composed of a host star, a transiting Jupiter-sized exoplanet, and an Earth-sized exoplanet. By performing a thorough and systematic orbit classification we map a large section of the phase space and determine the final states of the Earth-sized exoplanet. In particular, we manage to classify the initial conditions of the perturber into several categories such as bounded regular or chaotic motion, collision, and escaping motion. Bounded regular motion corresponds to regions of the phase space in which the orbit of the Earth-sized planet would be stable. In our computations, we use realistic values of the involved parameters and initial conditions corresponding to real observable exosystems. Our analysis provides important information on how the mass of the host star as well as the mass of the Jupiter size affects the orbital dynamics of the Earth-sized exoplanet.

A Stability Timescale for Nonhierarchical Three-body Systems

The gravitational three-body problem is a fundamental problem in physics and has significant applications to astronomy. Three-body configurations are often considered stable as long the system is hierarchical; that is, the two orbital distances are well-separated. However, instability, which is often associated with significant energy exchange between orbits, takes time to develop. Assuming two massive objects in a circular orbit and a test particle in an eccentric orbit, we develop an analytical formula estimating the time it takes for the test particle’s orbital energy to change by an order of itself. We show its consistency with results from N-body simulations. For eccentric orbits in particular, the instability is primarily driven not by close encounters of the test particle with one of the other bodies, but by the fundamental susceptibility of eccentric orbits to exchange energy at their periapsis. Motivated by recent suggestions that the galactic center may host an intermediate-mass black hole (IMBH) as a companion to the massive black hole Sgr A*, we use our timescale to explore the parameter space that could harbor an IMBH for the lifetime of the S-cluster of stars surrounding Sgr A*. Furthermore, we show that the orbit of an S-star can be stable for long timescales in the presence of other orbital crossing stars, thus suggesting that the S-cluster may be stable for the lifetimes of its member stars.

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.

Controlled short-period orbits around Earth-Moon equilateral libration points for Lunar Occultations

This paper proposes a framework for a satellite mission with the goal to perform Lunar Occultations within the context of the circular restricted three-body problem (CR3BP) of the Earth-Moon system. This is achieved by exploiting the geometrical relationships between the Earth, Moon, and a spacecraft orbiting around L4. Additionally, it examines the viability of a linear state-feedback control policy to stabilize short-period orbits under the presence of the gravitational effect of the Sun, Solar radiation pressure, and the Lunar orbital eccentricity, which is not accounted for in the CR3BP dynamics. Given the unprecedented NUV angular resolution achieved by a spacecraft at L4, this research proposes a framework for a remote sensing observatory for Solar activity, thereby enabling the monitoring of the entire hemisphere of Solar radiation, and to study different astrophysical issues via Lunar Occultations.

Framework for the full N-body problem in SE(3) and its reduction to the circular restricted full three-body problem

Framework for the full N-body problem in SE(3) and its reduction to the circular restricted full three-body problem

Dynamical Stability of Polar Circumbinary Orbits and Planet Formation in the Planetary Disk of 99 Herculis

A possible polar-ring debris disk, the dynamics of which can be described by the outer hierarchical restricted three-body problem, has been detected in 99 Herculis. An empirical formula for the minimum radius beyond which test particles in polar orbits can keep stable within 107 binary periods is provided through numerical fitting, applying to the binary eccentricity and the mass ratio of the binary , where λ = m 0/m 1 (m 0 and m 1 represent the masses of the two binary stars). The polar planetary disk has the lowest statistical accretion efficiency and a moderate impact frequency of collisions among planetesimals (with radii of 1–10 km) compared to those in a circumbinary coplanar disk and a standard disk around the single host star. The collision timescale in the circumbinary disk (both polar and coplanar configurations) is longer than 107 yr, exceeding the dissipation timescale of the gas disk. Stochastic simulations show that successive collisions cannot make planetesimals grow, which may explain the formation of the debris disk observed in 99 Herculis.

Powered lunar flyby for transfers between non-coplanar orbits around Earth

This work studies transfer between non-coplanar circular orbits around Earth with the space vehicle performing a powered lunar flyby maneuver. The complete transfer trajectory is accomplished by an application of two or three impulsive velocity increments. First and final velocity increments are applied tangentially, respectively, to the departing and the arrival orbits around Earth. An optional second velocity increment is applied at the perilune in order to increase the effects of the flyby maneuver. Despite many works consider the powered lunar flyby instead of a natural lunar flyby, it is important to compare both maneuvers in the context of the complete trajectory. In this direction, the present work formulates and solves multiple point boundary value problems that determine the transfer trajectories considering three models: a three-dimensional patched-conic approximation, a model based on the spatial restricted three-body problem, and, a model based on the spatial bi-circular restricted four-body in which the influence of the Sun is included. The transfer trajectory solutions are compared with classical maneuvers and with transfers that perform a natural flyby maneuver. An interesting result shows that a decelerating propulsion during the flyby maneuver can provide a transfer trajectory with a fuel consumption smaller than the one of bi-parabolic maneuver even if the Sun’s attraction is considered. Moreover, the influence of the Sun can decrease the time of flight and the apogee of the trajectory and it can save fuel consumption if the Sun’s initial phase angle is properly chosen.

On Resonant Values of Parameters in the Problem on the Stability of Lagrangian Solutions in the Near-Circular Restricted Three-Body Problem

The restricted problem of three bodies (material points) moving under the action of gravitational attraction according to Newton’s law is considered. The orbits of the main attracting bodies are considered to be ellipses with a small eccentricity, and a passively gravitating body can perform arbitrary spatial motion near the triangular libration point. For the Hamiltonian function corresponding to such a motion, the structure of the normal form is pointed in the cases of third-order resonances. In the planar restricted three-body problem, the equations up to the second degree of eccentricity for resonance curves for all resonances up to the sixth order inclusive are obtained.

AVERAGED CIRCULAR SPATIAL RESTRICTED THREE-BODY PROBLEM: INTERNAL CASE, NEW RESULTS

We investigate the spatial restricted circular three-body problem in the nonresonant case. Namely, we apply Gaussian averaging to obtain averaged equations in terms of osculating elements and then investigate them. Keplerian ellipse with a focus in the main body (the Sun) is taken as an unperturbed orbit assuming the semi-major axis of the ellipse to be less than the radius of the orbit of the outer planet (internal problem). Using the Parseval formula we have derived the twice-averaged perturbed force function of the problem in the form of an explicit analytical series with coefficients expressed in terms of the Gauss and Clausen hypergeometric functions. An investigation of averaged force function along it’s curves of non-analyticity showed that the series are asymptotic by Poincaré. For a reduced system with one degree of freedom, phase portraits of oscillations in the plane of the Keplerian elements \(e\), \(\omega \) are constructed in the second and fourth approximations. It is shown the topology of the phase portrait is more complicated in fourth approximation then in first and second approximations provided that the constant \({{c}_{1}}\) of the Lidov–Kozai integral belongs to the interval (0, 0.382).