Three-body Problem Research Articles

Partial suppression of chaos in relativistic three-body problems

Partial suppression of chaos in relativistic three-body problems

Newtonian restricted three-body gravitational problem with positive and negative point masses

The Newtonian restricted three-body problem involving a positive primary point mass, m+, and a negative secondary point mass, m−, in a circular orbit, and a positive or negative tertiary point mass, m3, with m+>|m−|≫|m3|, is solved. Five Lagrange points are found for m3, three of which are coplanar with m+ and m−, and two of which are not, a subtle consequence of the gravitational repulsion from m−. All Lagrange points are linearly unstable, except for one point in the regime m+≳8.4|m−|, which is linearly stable and collinear with m+ and m−. Published by the American Physical Society 2024

A dynamical study of Hilda asteroids in the Circular and Elliptic RTBP.

The Hilda group is a set of asteroids whose mean motion is in a 3:2 orbital resonance with Jupiter. In this paper, we use the planar Circular Restricted Three-Body Problem (CRTBP) as a dynamical model and we show that there exists a family of stable periodic orbits that are surrounded by islands of quasi-periodic motions. We have computed the frequencies of these quasi-periodic motions and we have shown how the Hilda family fits inside these islands. We have compared these results with the ones obtained using the Elliptic Restricted Three-Body Problem and they are similar, showing the suitability of the CRTBP model. It turns out that, to decide if a given asteroid belongs to the Hilda class, it is much better to look at its frequencies in the planar CRTBP rather than to use two-body orbital elements as it is commonly done today.

Study of the resonant motion of a test particle inside Kepler 69 using circular restricted three body problem

Summary: Scientific researches to discover extraterrestrial life are very intense and important. For a long time, humanity has speculated about the existence of the planetary systems different from our solar system, where extraterrestrial life can be existing elsewhere in the universe. The understanding of the origin of planets and planetary systems has indeed become a major focus of research in modern Astrophysics. Exoplanet research is one of the most field developing subjects in Astrophysics and Astronomy. Today, the number of discovered exoplanets, where life is thought to exist, is large. One of the main goals of scientists is to find Earth 2, where there is possibility of extraterrestrial life. Kepler's mission was, and is very important in the field of exoplanet research. This mission has revolutionized our understanding of exoplanets. The number of discovered exoplanets has exceeded 5780 in 4314 planetary systems, with 969 systems having more than one planet [1]. Among them, one of the most interesting is Kepler 69 c, a super-Earth around Kepler 69. This exoplanet is located at the boundary between the habitable zone and the outer one. In this study let’s focus on the classical gravitational problem called Circular Restricted Three-Body Problem (CRTBP) [2, 3] and use an iterative method to perform numerical calculations through MATLAB software. Therefore, in this paper, let’s try to study the movement of a “test particle”, under the action of the gravitational field of the system Kepler 69–Kepler 69 c. The aim of this paper is to study the motion of a “test particle” (for example an extrasolar asteroid test) inside the exoplanetary system Kepler 69. Results are presented in the (x, y) plane in the rotating and inertia system. Also, some of the results are presented in the phase planes (x, vx) and (y, vy)

Bifurcation analysis of figure-eight choreography in the three-body problem based on crystallographic point groups

Abstract The bifurcation of figure-eight choreography is analyzed by its symmetry group based on the variational principle of the action. The irreducible representations determine the symmetry and the dimension of the 
Lyapunov-Schmidt reduced action, which yields four types of bifurcations in the sequence of the bifurcation cascade. Type 1 bifurcation, represented by trivial representation, bifurcates two solutions. Type 2, by non-trivial one-dimensional representation, bifurcates two congruent solutions. Type 3 and 4, by two-dimensional irreducible representations, bifurcate two sets of three and six congruent solutions, respectively. We analyze numerical bifurcation solutions previously published and four new ones: 
non-symmetric choreographic solution of type 2, non-planar solution of type 2, y-axis symmetric solution of type 3, and non-symmetric solution of type 4.

Analysis of radiation pressure and albedo effect in the generalized CR3BP with oblateness

This research investigates the impact of radiation pressure, albedo effect, and oblateness in the generalized Circular Restricted Three-Body Problem (CR3BP). We have considered the bigger primary is a radiating primary and the smaller primary produces the albedo effect. Moreover, the smaller primary is also considered as oblate body with zonal harmonic coefficient Ji;i=2,4. In this study, we have computed the equilibrium points and analyzed their stability, examined zero velocity curves to determine regions of possible motion, and described the periodic orbits around the equilibrium points. We observed that the equilibrium points are more significantly influenced by radiation pressure compared to the albedo effect. Furthermore, larger values of the oblateness parameters J2 and J4 notably impact on the equilibrium points. Moreover, an analysis is performed using the radiation pressure and albedo effect to assess the linear stability of all equilibrium points, and it is found that the collinear points are unstable, whereas, non-collinear equilibrium points are stable. We have obtained periodic orbits around the equilibrium points and observed that the amplitudes in the y− axes of the periodic orbit around L1 and L2 affected due to consideration of these perturbations, whereas the orbit around L3 is not affected by any of these considered perturbations. This research provides a comprehensive analysis of these perturbations, offering new perspectives on their roles in the generalized CR3BP.

General Three-Body Problem in Conformal-Euclidean Space: New Properties of a Low-Dimensional Dynamical System

Despite the huge number of studies of the three-body problem in physics and mathematics, the study of this problem remains relevant due to both its wide practical application and taking into account its fundamental importance for the theory of dynamical systems. In addition, one often has to answer the cognitive question: is irreversibility fundamental for the description of the classical world? To answer this question, we considered a reference classical dynamical system, the general three-body problem, formulating it in conformal Euclidean space and rigorously proving its equivalence to the Newtonian three-body problem. It has been proven that a curved configuration space with a local coordinate system reveals new hidden symmetries of the internal motion of a dynamical system, which makes it possible to reduce the problem to a sixth-order system instead of the eighth order. An important consequence of the developed representation is that the chronologizing parameter of the motion of a system of bodies, which we call internal time, differs significantly from ordinary time in its properties. In particular, it more accurately describes the irreversible nature of multichannel scattering in a three-body system and other chaotic properties of a dynamical system. The paper derives an equation describing the evolution of the flow of geodesic trajectories, with the help of which the entropy of the system is constructed. New criteria for assessing the complexity of a low-dimensional dynamical system and the dimension of stochastic fractal structures arising in three-dimensional space are obtained. An effective mathematical algorithm is developed for the numerical simulation of the general three-body problem, which is traditionally a difficult-to-solve system of stiff ordinary differential equations.

Cantor set structure of the weak stability boundary for infinitely many cycles in the restricted three-body problem

Cantor set structure of the weak stability boundary for infinitely many cycles in the restricted three-body problem

Sun–Jupiter–Saturn System May Exist: A Verified Computation of Quasiperiodic Solutions for the Planar Three-Body Problem

In this paper, we present evidence of the stability of a model of our Solar System when taking into account the two biggest planets, a planar (Newtonian) Sun–Jupiter–Saturn system with realistic data: masses of the Sun and the planets, their semiaxes, eccentricities and (apsidal) precessions of the planets close to the real ones. (We emphasize that our system is not in the perturbative regime but for fixed parameters.) The evidence is based on convincing numerics that a KAM theorem can be applied to the Hamiltonian equations of the model to produce quasiperiodic motion (on an invariant torus) with the appropriate frequencies. To do so, we first use KAM numerical schemes to compute translated tori to continue from the Kepler approximation (two uncoupled two-body problems) up to the actual Hamiltonian of the system, for which the translated torus is an invariant torus. Second, we use KAM numerical schemes for invariant tori to refine the solution giving the desired torus. Lastly, the convergence of the KAM scheme for the invariant torus is (numerically) checked by applying several times a KAM–iterative lemma, from which we obtain that the final torus (numerically) satisfies the existence conditions given by a KAM theorem.

The Three-Body and n-Body Problems

This project explores the multi-body problem according to Newton's Laws of Gravitation in two dimensions. First, I will derive and numerically solve the differential equations which govern the motion of the three-body problem. Then, I will explore the chaotic nature of the system through a variety of demonstrations, plots, animations, and Explore windows. After, I will look at a few special periodic solutions to the three-body problem. Next, I will apply this system to astronomy, exploring the gravitational orbits that comprise our solar system. Finally, I will to generalize the three-body problem into a higher order n-body system (specifically, five bodies).

Regular Quaternion Equations of the Spatial Hill Problem in Kustaanheimo–Stiefel Variables and Quaternion Osculating Elements

Regular quaternion equations of the spatial Hill problem (a variant of the limited three-body problem (Sun, Earth, Moon (or another low-mass moving cosmic body under study)) are obtained, when the distance between two bodies with finite masses is considered very large, in four-dimensional Kustaanheimo-Stiefel variables (KS-variables) within the framework of the elliptical and circular spatial bounded three-body problem, as well as the regular quaternion equations of the planar Hill problem in two-dimensional Levi-Civita variables. In these equations, the variables are KS-variables or Levi-Civita variables and the energy of relative motion of the body under study, or a variable that converts for the circular Hill problem into a constant of motion of this body (the Jacobi integration constant), as well as the planetocentric distance of the Sun and real time associated with a new independent variable by the Sundman differential transformation of time or other more complex differential ratio. These equations are supplemented by the equation of the Earth’s orbit in polar coordinates and the equation for the true anomaly characterizing the Earth’s position in the orbit. The first integral of the obtained equations in KS-variables in the case of a circular problem is established. Another first partial integral in the general case is a bilinear relation connecting KS-variables and their first derivatives. Three new forms of regular equations of the spatial Hill problem in quaternion osculating elements (slowly changing quaternion variables) are proposed. The proposed regular quaternion equations have an oscillatory form or the form of equations with slowly changing variables, which makes it possible to effectively use analytical and numerical methods of oscillation theory and methods of nonlinear mechanics in the study of the Hill problem.

Feasibility of orbital capture of near-earth asteroids based on the planar-circular restricted three-body problem

With the goal of efficiently extracting samples or even materials from the surface of an asteroid, this study proposed and investigated a method to change the velocity vector of a near-Earth asteroid and place it into an orbit where it is captured by the Earth’s gravitational field. The change in the orbit of an asteroid is not directly discussed in relation to the change in the velocity vector but is indirectly considered by the change in the Jacobi integral, which is the first integral of the circular-planar restricted three-body problem. In addition, the distribution of the smaller alignment index (SALI) is investigated to find a capture point where the asteroid is not put into a chaotic orbit. The proposed method is numerically demonstrated for fictional asteroid capture missions. The results show that several asteroids can be put into stable captured orbits. Additionally, we propose a method to optimize the value of the Jacobi integral, aiming to stabilize periodic captured orbits. Numerical integration confirms that when the Jacobi integral is optimized, the orbital lifetime of the captured orbit exceeds 500 years.

Optimization of tethered artificial gravity assists for capture about binary asteroids in the circular restricted three-body problem

The use of tethered artificial gravity assists in space travel to perform orbital maneuvers has the potential to significantly extend missions through the reduction of fuel consumption. As space exploration in the solar system has become more accessible, interest in near-Earth asteroid (NEA) missions has grown. Studying NEAs can yield new information about the early solar system as well as provide important insights into related fields, such as planetary defense. Further, it has been discovered that many NEAs are in binary systems, which presents a unique orbital environment for optimizing spacecraft maneuvers for future asteroid missions. This paper develops an approach employing a genetic algorithm to determine optimal tethered maneuvers to periodic orbits in the vicinity of a binary asteroid using circular restricted three-body problem (CR3BP) dynamics. The optimization identifies the maneuver that results in the greatest change in the Jacobi constant values between the incoming orbit and final desired orbit. This maximizes the benefit of the tethered artificial gravity assist into a periodic orbit about the primary asteroid. The tethered gravity assists are examined for binary asteroid systems with various mass ratios using CR3BP dynamics. This investigation selected three observed binary asteroid systems with orbital eccentricities that allow for modelling using the CR3BP. Optimizations were performed and simulated for the three systems with different mass ratios to investigate and demonstrate the viability of tethered gravity assists in various binary asteroid environments.

Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem

In this paper, we intend to investigate the dynamics of the Circular Restricted Three-Body Problem. Here we assumed the primaries as the source of radiation and have variable mass. The gravitational perturbation from disk-like structure are also considered in this study. There exist five equilibrium points in this system. By considering the combined effect of disk–like structure and the mass transfer, we found that the classical collinear equilibrium points depart from the x–axis. We called these equilibrium points as quasi–collinear equilibrium points. Meanwhile, this combined effect also breaks the symmetry of triangular equilibrium point positions. We noted that the quasi–equilibrium points are unstable whereas the triangular equilibrium points are stable if the mass ratio μ is smaller than critical mass μc. Besides the mass ratio, the stability of triangular equilibrium points depend on time.

Equilibrium points and locations in the photogravitational circular Robe’s restricted three-body problem with variable masses

The restricted three-body problem (R3BP) defines the dynamics of an infinitesimal mass moving in the gravitational neighborhood of two primaries, which move in circular orbits around their center of mass on account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries. In this paper, we examine equilibrium points and their locations in the photogravitational circular Robe’s restricted three-body problem (R3BP) with variable masses. The motion of the primaries and variation in masses of the primaries are governed by the Gylden-Mestschersky problem (GMP) and the unified Mestschersky law (UML), respectively, while the second primary is assumed to be a radiation emitter. The non-autonomous equations of the governing dynamical system are deduced and transformed using the Mestschersky transformation (MT), the UML and the particular solutions of the GMP, to a system of the autonomized equations with constant coefficients under the condition that there is no fluid inside the first primary. Next, the equilibrium points (EPs) of the autonomized system are explored using perturbation method and it is seen that axial EP which is defined by the mass parameter and the radiation pressure of the second primary exists. Further, a pair of non-collinear EPs which depends on the mass parameter, radiation pressure of the second primary and a constant of the mass variations of the primaries, is found. The EPs of the non-autonomous system are obtained using the MT and differ from those of the autonomized system by time t . The EPs may be used in different problems of stellar dynamics, and also in other astrophysical applications.

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.

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.

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.

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.