Three-body Problem Research Articles

Kepler’s problem of a two-body system perturbed by a third body

One of the most important problems in basic physics and astronomy is studying the motion of planets, satellites, and other celestial bodies. The solution to the two-body problem enables astronomers to predict the orbits of the Moon, satellites, and spaceships around the Earth. The general analytic solution for the three-body problem stands unsolved except in some special cases. This reduces the problem to a two-body problem. In this work, the authors present a closed-form approach to the three-body problem theoretically and numerically based on particle–particle vector analysis. The theoretical approach, which is based on the real Moon–Sun–Earth problem information, illustrates the perturbation of the Moon in the Sun–Earth problem and shows an expected orbital motion with a perturbation in the Sun–Earth orbit due to the revolution of the Moon. The numerical investigation uses the same information to study the same problem and calculate the angular momentums of each pair of objects. The two solutions show good agreement with the well-known Earth-Moon and Sun–Earth momentums. The Moon–Sun orbit is close to an elliptic shape with angular momentum of about 3.27 × 1038 J.s. This approach is the key to future studies for n-body problem solutions.

Out-of-Plane Equilibrium Points in the Photogravitational Hill Three-Body Problem

This paper investigates the movement of a negligible mass body (third body) in the vicinity of the out-of-plane equilibrium points of the Hill three-body problem under the effect of radiation pressure of the primaries. We study the effect of the radiation parameters through the factors qi,i=1,2 on the existence, position, zero-velocity curves and stability of the out-of-plane equilibrium points. These equilibrium positions are derived analytically under the action of radiation pressure exerted by the radiating primary bodies. We determined that these points emerge in symmetrical pairs, and based on the values of the radiation parameters, there may be two along the Oz axis and either none or two on the Oxz plane (outside the axes). A thorough numerical investigation found that both radiation factors have a strong influence on the position of the out-of-plane equilibrium points. Our results also reveal that the parameters have impact on the geometry of the zero-velocity curves. Furthermore, the stability of these points is examined in the linear sense. To do so, the spatial distribution of the eigenvalues on the complex plane of the linearized system is visualized for a wide range of radiation parameter combinations. By a numerical investigation, it is found that all equilibrium points are unstable in general.

Tisserand-Leveraging Lambert Problem: Formulation, Solution, and Applications

Tisserand-leveraging transfers (TILTs) were proposed as techniques to achieve an endgame tour with small orbit insertion maneuvers and short flight times in low-energy regions of the planar, circular, restricted three-body problem. This paper presents a new formulation of TILTs with a framework similar to the classical Lambert problem. The developed algorithm searches for trajectories of TILTs with constraints on boundary values and time of flight by solving a one-dimensional root-finding problem. Compared to the traditional algorithm for TILTs, the developed algorithm enhances its generality by removing the restriction on the initial and final phases of the transfer. Additionally, the algorithm can achieve trajectory patching that includes TILTs without reoptimization and tour redesign based on the actual position of the spacecraft due to the Lambert-like input–output design. The algorithm uses existing neural network tools for fast and broad search in the solution space. Simulation results indicate that the algorithm can explicitly find TILT solutions and families satisfying boundary value constraints. Furthermore, the algorithm is applied to design a new endgame trajectory from Ganymede to Europa, demonstrating the potential of integrating the new algorithms with other trajectory design tools.

Semi-analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problem

A unified semi-analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits, quasihalo orbits, Axial orbits, and their invariant manifolds, as well as transit and non-transit orbits. Based on classical in-plane and out-of-plane frequency resonance mechanisms, the Lindstedt–Poincaré method could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. In this paper, by introducing a coupling coefficient η and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt–Poincaré method. Analyzing the bifurcation equation obtained from different coupling forms reveals bifurcation conditions for all kinds of orbits near collinear libration points. When η = 0, the series solution describes non-bifurcated orbits, while when η ≠ 0, the solution describes bifurcated orbits, including quasihalo orbits, Axial orbits, and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified semi-analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.

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.

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.

Perturbing and Oblateness's Impact on the Generalized Photogravitational Restricted Three-Body Problem

In this research work, the effects of perturbing and oblateness on the generalised photogravitational restricted three body problem (RTBP) are examined when the primary are regarded as radiation sources and oblate spheroid. The perturbing forces were included in the equations of motion that we developed for the system. The triangular libration points were derived in terms of the perturbing parameters using linear approximations, and their influence was seen. Applying Murray's criteria, we determined that the triangular libration points remained unstable due to the nature of the characteristics equation matching to the variational equations of motion of the system generated.

The Schubart Orbits in the Curved Three-Body Problem with Two Equal Masses

The Schubart Orbits in the Curved Three-Body Problem with Two Equal Masses

A Digital Health Humanities Approach to the Three-Body Problem in Healthcare Education: Balancing the Science, Art and Tech with M.A.G.I.C.

In the provision of patient-centered care, healthcare professionals face a challenging triad that is analogous to the classical "Three-Body Problem" in physics, whereby any shifts in the dynamics of medical knowledge, narrative competence and digital technologies can tip the balance of healthcare education and practice. Drugs/Medications are often caught in the dichotomy of being both "heroes" and "villains", which underscores the necessity of a balanced, informed, and empathetic approach to patient care. Drawing inspiration from the popular novel and Netflix series, the dynamic interplay between the science of medical knowledge, the art of empathy and narrative competence, and the advancement of healthcare technologies to harmonise the diverse, yet interconnected domains, of medication management, medical humanities and digital health is demonstrated - in the emerging field of Digital Health Humanities. The novel project called M.A.G.I.C. is revealed, that blends the rigour of medical science with the nuanced understanding of patient narratives, and the cutting-edge potential of digital innovations like Generative Artificial Intelligence and the Metaverse, to create a holistic, learner-centered approach for the education of the healthcare professions.

Celestial Mechanics: The Non-Stability of The Newtonian Model

Multiple researchers and the three-body problem highlight a potential instability in the Classic model of the Solar System, indicating a possible fundamental instability in the system's dynamics. Therefore, we consider that the Classical model does not reflect reality, and we propose an alternative model that we call: Theory of Dynamic Interactions, providing a brief exposition of the studies carried out by Advanced Dynamics, until defining a Rotational Dynamics of Interactions, applicable to bodies subjected to multiple pairs of successive non-coaxial forces.

Hierarchical Three-Body Problem at High Eccentricities = Simple Pendulum II: Octupole including Brown’s Hamiltonian

Abstract The very long-term evolution of the hierarchical restricted three-body problem with a massive perturber is analyzed analytically in the high eccentricity regime. Perturbations on the time scale of the outer orbit can accumulate over long timescales and be comparable to the effect of the octupole term. These perturbations are described by Brown’s Hamiltonian - having different forms in the literature. We show that at the high eccentricity regime - the effect of Brown’s Hamiltonian is an azimuthal precesssion of the eccentricity vector and can be solved analytically. In fact, the dynamics are equivalent to a simple pendulum model allowing an explicit flip criterion.

Λ6He (1−, 2−) spin doublet in a Λ + n + α cluster model

The spin singlet and spin triplet states of the helium-6-lambda hypernucleus are studied in configuration space as a three-body problem, using the hyperspherical method. This hypernucleus is treated as a Λ + n + α cluster model. Since the α particle core has a spin of 0, this is an ideal hypernucleus from which to understand spin dependence in hyperon-nucleon potentials. In this paper, the hyperon-nucleon potential considered is the GLM-YN0 Λ-n potential, which is a simulation of the NSC97f potential obtained through inverse scattering theory. Due to violations of the cluster structure, two-body potentials underbind the helium-6-lambda hypernucleus. This issue is addressed by including a three-body force to complement these two-body forces. A three-body force, whose strength is spin dependent, is used to guide predictions of the spin triplet state of helium-6-lambda which is not yet known experimentally.

Sundman theorem revisited

In this paper, we revisit Sundman’s Theorem on the three-body problem. We provide a simple example demonstrating that the requirement on the total angular momentum L (bounded away from zero) can be relaxed for a particular set of initial conditions, allowing double shocks. We present an explicit solution for the exact triple-collision scenario through a Sundman-like formal regularization.

An efficient approach for searching three-body periodic orbits passing through Eulerian configuration

A new efficient approach for searching three-body periodic equal-mass collisionless orbits passing through Eulerian configuration is presented. The approach is based on a symmetry property of the solutions at the half period. Depending on two previously established symmetry types on the shape sphere, each solution is presented by one or two distinct initial conditions (one or two points in the search domain). A numerical search based on Newton’s method on a relatively coarse search grid for solutions with relatively small scale-invariant periods T∗<70 is conducted. The linear systems at each Newton’s iteration are computed by high order high precision Taylor series method. The search produced 12,431 initial conditions (i.c.s) corresponding to 6333 distinct solutions. In addition to passing through the Eulerian configuration, 35 of the solutions are also free-fall ones. Although most of the found solutions are new, all linearly stable solutions among them (only 7) are old ones. Particular attention is paid to the details of the high precision computations and the analysis of accuracy. All i.c.s are given with 100 correct digits.

A search for Planet Nine with small spacecraft: Three-body, post-Newtonian, non-gravitational, planetary and Kuiper Belt effects

A hypothetical gravitating body in the outer Solar System, the so-called Planet Nine, was proposed to explain the unexpected clustering of the Kuiper Belt Objects. As it has not been observed via telescopes, it was conjectured to be a primordial black hole (of the size of a quince) that could be gravitationally detected by laser-launching or solar sailing many small spacecraft. Here, we study various aspects that will affect such a search for Planet Nine. Our basic observable is the angular displacement in the trajectory of a small spacecraft which will be mainly affected by the gravity of Planet Nine, augmented with several other 3-body, non-gravitational, post-Newtonian, planetary and Kuiper Belt effects. First, we calculate the effect of the Sun in the framework of the circular restricted three-body problem of the Sun–Planet Nine-spacecraft for the two particular initial conditions. Then, we study the effects of Kuiper Belt and outer planets, namely Jupiter, Saturn, Uranus, Neptune, as well as non-gravitational perturbations such as magnetic and drag forces exerted by the interstellar medium; and the solar radiation pressure. In addition, we investigate the post-Newtonian general relativistic effects such as the frame-dragging, Schwarzschild effect, and geodetic precession on the spacecraft trajectory. We show that the leading order angular displacement is due to the solar radiation pressure for the lower spacecraft velocities, and the drag force for the higher spacecraft velocities. Among the general relativistic effects, the frame-dragging has the smallest effect; and the Schwarzschild effect due to Sun has the largest effect. However, none of the general relativistic effects produces a meaningful contribution to the detection.

Accurate Simulation of Efimov Physics in Ultracold Atomic Gases with Realistic Three-Body Multichannel Interactions

We give a detailed and self-contained description of a recently developed theoretical and numerical method for the simulation of three identical bosonic alkali-metal atoms near a Feshbach resonance, where the Efimov effect is induced. The method is based on a direct construction of the off-shell two-body transition matrix from exact eigenfunctions of the embedded two-body Hamiltonians, obtained using realistic parameterizations of the interaction potentials which accurately reproduce the molecular energy levels. The transition matrix is then inserted into the appropriate three-body integral equations, which may be efficiently solved on a computer. We focus especially on the power of our method in including rigorously the effects of multichannel physics on the three-body problem, which are usually accounted for only by various approximations. We demonstrate the method for 7Li, where we recently showed that a correct inclusion of this multichannel physics resolves the long-standing disagreement between theory and experiment regarding the Efimovian three-body parameter. We analyze the Efimovian enhancement of the three-body recombination rate on both sides of the Feshbach resonance, revealing strong sensitivity to the spin structure of the model thus indicating the prevalence of three-body spin-exchange physics. Finally, we discuss an extension of our methodology to the calculation of three-body bound-state energies.

Investigating temporary capture in the Sun-Jupiter three-body system via Lagrangian coherent structures

Abstract The temporary capture (TC) of Jupiter-family objects has long been a pivotal focus in celestial mechanics research. This study investigates the temporary capture of objects near Jupiter within the context of the planar circular restricted three-body problem (PCRTBP), employing Lagrangian coherent structures (LCSs) and periapsis Poincaré maps. Initially, LCSs are identified via periapsis Poincaré maps and applied to segment the phase space. Parameter scanning enables a detailed analysis, classifying the orbital behaviors of objects in the proximity of Jupiter into three distinct categories: temporary capture, low-energy flyby, and collision, each designating specific regions in phase space. Subsequently, a novel method for screening potential TC objects within the Jupiter system is proposed and validated, informed by the dynamic characteristics of TC motions. The efficacy of this method is illustrated by the reidentification of six known TC comets and the prediction of a prospective TC asteroid, 2002 GV28. Within the framework of the PCRTBP, analogous TC trajectories for these comets and asteroids are identified, offering novel insights into the dynamics of temporary capture events.

Investigation of motion around out-of-plane points in the restricted three-body problem with variable shape and masses

The paper examines out-of-plane equilibrium points (OEPs) of the restricted three-body problem with variable masses and shape. The bigger primary varies it shape as the lengths of the semi-axes vary with time. For the autonomized system, two pair of OEPs L6,7κ and L8,9κ, are obtained and differ from those of the non-autonomous system due to time t. The stability of OEPs of both systems is found to be unstable. Further, numerical illustrations is provided when variations in shape of the bigger primary is, a triaxial prolate, a sphere and a triaxial oblate shape. The positions, stability and zero velocity curves (ZVC) of the particle around the OEPs are explored. It is seen that when the bigger primary is a triaxial prolate body, the OEPs L8,9κ are closer to the primaries than L6,7κ. However, the converse happens when it is a triaxial oblate body. Also, when the bigger primary is a triaxial oblate body, the OEPs are farther away from the primaries than when it is a triaxial prolate. In the case of the ZVC, it is seen that when the bigger primary is a triaxial prolate body, there is a petal around it, and region of allowed motion of the particle increases, while the region reduces when the bigger primary evolves from a sphere to a triaxial oblate body. This study can be used to describe motion of a dust grain in the vicinity of Betelgeuse, a red giant star whose mass and shape changes with time and its stellar companion.

Parametric approximations of fast close encounters of the planar three-body problem as arcs of a focus-focus dynamics

Abstract A gravitational close encounter of a small body with a planet may produce a substantial change of its orbital parameters which can be studied using the circular restricted three-body problem. In this paper we provide parametric representations of the fast close encounters with the secondary body of the planar CRTBP as arcs of non-linear focus-focus dynamics. The result is the consequence of a remarkable factorisation of the Birkhoff normal forms of the Hamiltonian of the problem represented with the Levi–Civita regularisation. The parameterisations are computed using two different sequences of Birkhoff normalisations of given order N. For each value of N, the Birkhoff normalisations and the parameters of the focus-focus dynamics are represented by polynomials whose coefficients can be computed iteratively with a computer algebra system; no quadratures, such as those needed to compute action-angle variables of resonant normal forms, are needed. We also provide some numerical demonstrations of the method for values of the mass parameter representative of the Sun–Earth and the Sun–Jupiter cases.

Family of 2:1 resonant quasi-periodic distant retrograde orbits in cislunar space

Given the current enthusiasm for lunar exploration, the 2:1 resonant distant retrograde orbit (DRO) in Earth-Moon space is of interest. To gain an in-depth understanding of the complex dynamic environment in cislunar space and provide more options for parking orbits, this paper investigates the existence of quasi-periodic orbits near the 2:1 resonant DRO in the circular restricted three-body problem (CR3BP). Firstly, the numerical computation approach, continuation strategy, and stability analysis method of quasi-periodic orbits are introduced. Then, addressing the primary challenges in the continuation progress, we have developed an adaptive continuation algorithm with automatic adjustment of the step size and the number of discrete points that characterize the invariant torus. Finally, two types of 2D quasi-DROs and their linear stability properties are explored. Using Poincaré sections, we investigated the boundaries of the maximum extent attainable by both 2D quasi-DRO families in the CR3BP at a specific Jacobi energy, confirming that both types of quasi-periodic families have reached their respective boundaries. The algorithm described in this paper is beneficial for facilitating the computation of quasi-periodic families and aids in discovering additional potential dynamical structures.