General Three-body Problem Research Articles

On an Application of the Hill Approach to the General Case of the Three-body Problem

In this work, we use the energy and angular momentum integrals as a resource for applying Hill’s approach to the general three-body problem. As a result, we obtain theorems on the Lagrange stability and Hill stability in the general three-body problem. Also, specific features of the general and restricted three-body problems are discussed.

Non-linear dynamics of a test particle near the Lagrange points L4 and L5 (Earth-Moon and Sun-Earth case)

The two-bodies problem can be fully solved, and was solved by Kepler (1609) and Newton (1687). The general three-body problem is often given as an example of a mathematical problem that ‘can’t be solved’. So, there is no general analytical solution. This problem can be significant and a special case of this problem is the Circular Restricted Three-Body Problem (CRTBP), which can be applied to the Earth-Moon system with a spacecraft, the Sun-Earth system with an asteroid, etc. In this paper, let’s focus on the motion of a test particle near the triangular Lagrange points L4 and L5 in the Earth-Moon and the Sun-Earth systems. Studying the movement of an object around these points is especially important for space mission design. To generate a trajectory around these points, the non-linear equations of motion for the circular restricted three-body problem were numerically integrated into MATLAB® 2023 software and the results are presented in the plane (x, y) and the phase plane (x, vx) and (y, vy). By numerical orbit integration, it is possible to investigate what happens when the displacement is relatively large or short from the Lagrange points. Then the small astronomical body may vibrate around these points. The results in this paper are shown in the rotating and inertia axes. Various initial positions near the Lagrange points and velocities are used to produce various paths the test particle can take. The same examples of numerical studies of trajectories associated with Lagrange points are shown in the inertial and the rotating coordinates system and are discussed. From the results of the numerical tests performed in MATLAB® 2023, it is possible to saw that there are different types of periodic, quasi-periodic, and chaotic orbits

Equilibria in an accelerated three-body problem

The general three-body problem is investigated with the addition of a fixed external force applied to one of the masses. It is firstly demonstrated that the centre-of-mass of the three-body system accelerates. Then, it is demonstrated that only a single, unstable, collinear equilibrium solution exits in the accelerating frame. While unstable, it is also demonstrated that this single equilibrium configuration is in principle controllable using additional control accelerations distributed between the masses. Potential applications of such an accelerated collinear equilibrium configuration are discussed for the active manoeuvring of chains of small asteroids for space resource utilisation.

Zero velocity surface in the general three-body-problem

The zero-velocity surfaces of the general planar three-body problem are constructed in the form space, the factor-space of the configuration space by transfer and rotation. Such a space is the space of congruent triangles, and the sphere in this space is similar triangles. The integral of energy in the form space gives the equation of the zero velocity surface. These surfaces can also be obtained based on the Sundman inequality. Such surfaces separate areas of possible movement from areas where movement is impossible. Without loss of generality, we can assume that the constant energy is −1/2 and the sought for surfaces depend only on the magnitude of the angular momentum of the problem, J. Depending on this value, five topologically different types of surfaces can be distinguished. For small J, the surface consists of two separate surfaces, internal and external ones, the motion is possible only between them. With J increasing the inner surface increases, the outer surface decreases, the surfaces first have a common point at some value of J, with a further increasing of J, their topological type changes and finally the zero-velocity surface splits into three disjoint surfaces, the motion is possible only inside them. Examples of the corresponding surfaces are given for each of these types, their sections in the plane xy and in the plane xz and the surfaces themselves are constructed, their properties are studied.

Creep tide model for the three-body problem

We present a tidal model for treating the rotational evolution in the general three-body problem with arbitrary viscosities, in which all the masses are considered to be extended and all the tidal interactions between pairs are taken into account. Based on the creep tide theory, we present a set of differential equations that describes the rotational evolution of each body, in a formalism that is easily extensible to the N tidally interacting body problem. We apply our model to the case of a circumbinary planet and use a Kepler-38 like binary system as a working example. We find that, in this low planetary eccentricity case, the most likely final stationary rotation state is the 1:1 spin–orbit resonance, considering an arbitrary planetary viscosity inside the estimated range for the Solar System planets. The timescales for reaching the equilibrium state are expected to be approximately millions of years for stiff bodies but can be longer than the age of the system for planets with a large gaseous component. We derive analytical expressions for the mean rotational stationary state, based on high-order power series of the ratio of the semimajor axes a1∕a2 and low-order expansions of the eccentricities. These are found to very accurately reproduce the mean behaviour of the low-eccentric numerical integrations for arbitrary planetary relaxation factors, and up to a1∕a2 ~ 0.4. Our analytical model is used to predict the stationary rotation of the Kepler circumbinary planets and we find that most of them are probably rotating in a subsynchronous state, although the synchrony shift is much less important than our previous estimations. We present a comparison of our results with those obtained with the Constant Time Lag and find that, as opposed to the assumptions in our previous works, the cross torques have a non-negligible net secular contribution, and must be taken into account when computing the tides over each body in an N-extended-body system from an arbitrary reference frame. These torques are naturally taken into account in the creep theory. In addition to this, the latter formalism considers more realistic rheology that proved to reduce to the Constant Time Lag model in the gaseous limit and also allows several additional relevant physical phenomena to be studied.

On the Lagrange Stability of the Motion in the Spatial Elliptic Restricted Three-body Problem

Abstract We study the Lagrange stability in the spatial elliptic restricted three-body problem. In this study, we essentially use the boundedness of the motion of a small particle with respect to planar coordinates and show that this plays a key role in ensuring the boundedness of the motion in the spatial case. As a result, we obtain the Lagrange stability theorem for an infinitely small particle in the spatial elliptic restricted three-body problem. The proposed approach can then be extended to the case of the general three-body problem.

Exploiting periodic orbits as dynamical clues for Kepler and K2 systems

Aims. Many extrasolar systems possessing planets in mean-motion resonance or resonant chain have been discovered to date. The transit method coupled with transit timing variation analysis provides an insight into the physical and orbital parameters of the systems, but suffers from observational limitations. When a (near-)resonant planetary system resides in the dynamical neighbourhood of a stable periodic orbit, its long-term stability, and thus survival, can be guaranteed. We use the intrinsic property of the periodic orbits, namely their linear horizontal and vertical stability, to validate or further constrain the orbital elements of detected two-planet systems. Methods. We computed the families of periodic orbits in the general three-body problem for several two-planet Kepler and K2 systems. The dynamical neighbourhood of the systems is unveiled with maps of dynamical stability. Results. Additional validations or constraints on the orbital elements of K2-21, K2-24, Kepler-9, and (non-coplanar) Kepler-108 near-resonant systems were achieved. While a mean-motion resonance locking protects the long-term evolution of the systems K2-21 and K2-24, such a resonant evolution is not possible for the Kepler-9 system, whose stability is maintained through an apsidal anti-alignment. For the Kepler-108 system, we find that the stability of its mutually inclined planets could be justified either solely by a mean-motion resonance, or in tandem with an inclination-type resonance. Going forward, dynamical analyses based on periodic orbits could yield better constrained orbital elements of near-resonant extrasolar systems when performed in parallel to the fitting of the observational data.

Chaos and Lévy flights in the three-body problem

ABSTRACT We study chaos and Lévy flights in the general gravitational three-body problem. We introduce new metrics to characterize the time evolution and final lifetime distributions, namely Scramble Density $\mathcal {S}$ and the Lévy flight (LF) index $\mathcal {L}$, that are derived from the Agekyan–Anosova maps and homology radius $R_{\mathcal {H}}$. Based on these metrics, we develop detailed procedures to isolate the ergodic interactions and Lévy flight interactions. This enables us to study the three-body lifetime distribution in more detail by decomposing it into the individual distributions from the different kinds of interactions. We observe that ergodic interactions follow an exponential decay distribution similar to that of radioactive decay. Meanwhile, Lévy flight interactions follow a power-law distribution. Lévy flights in fact dominate the tail of the general three-body lifetime distribution, providing conclusive evidence for the speculated connection between power-law tails and Lévy flight interactions. We propose a new physically motivated model for the lifetime distribution of three-body systems and discuss how it can be used to extract information about the underlying ergodic and Lévy flight interactions. We discuss ejection probabilities in three-body systems in the ergodic limit and compare it to previous ergodic formalisms. We introduce a novel mechanism for a three-body relaxation process and discuss its relevance in general three-body systems.

On an analogy in the restricted and general three-body problems

It is shown that the approach proposed by the author in Sosnitskii (2017) to investigate the Lagrange stability of motions in the planar restricted three-body problem can also be applied to the planar case of the general three-body problem. In particular, we prove the Lagrange stability theorem, which is an analogue of Theorem 2 from Sosnitskii (2017).

Totally Conservative Integration Method for the General Three-body Problem and Its Lagrangian Solutions

Abstract In this paper, we design a precise integration method with a variable time step for the general three-body problem that maintains all the conserved quantities. Our method is based on a logarithmic Hamiltonian leapfrog with chain vectors proposed by Mikkola & Tanikawa and features an energy-preserving parameter. Although the proposed method is merely second-order accurate, it can precisely trace some periodic orbits. This is not possible with generic geometric eighth-order numerical integrators and the logarithmic Hamiltonian leapfrog approach. Further, similar to logarithmic Hamiltonian leapfrog, our method is analytically shown to have Lagrangian solutions. Prior to the presented integration method, no integration method was known to preserve all the conserved quantities, in addition to presenting triangular Lagrangian solutions. Because our method is implicit, it requires an iteration method. Therefore, the proposed approach seems to be computationally intensive. However, our method is less computationally burdensome than a generic explicit eighth-order symplectic method.

On quasi-satellite periodic motion in asteroid and planetary dynamics

Applying the method of analytical continuation of periodic orbits, we study quasi-satellite motion in the framework of the three-body problem. In the simplest, yet not trivial model, namely the planar circular restricted problem, it is known that quasi-satellite motion is associated with a family of periodic solutions, called family $f$, which consists of 1:1 resonant retrograde orbits. In our study, we determine the critical orbits of family $f$ that are continued both in the elliptic and in the spatial model and compute the corresponding families that are generated and consist the backbone of the quasi-satellite regime in the restricted model. Then, we show the continuation of these families in the general three-body problem, we verify and explain previous computations and show the existence of a new family of spatial orbits. The linear stability of periodic orbits is also studied. Stable periodic orbits unravel regimes of regular motion in phase space where 1:1 resonant angles librate. Such regimes, which exist even for high eccentricities and inclinations, may consist dynamical regions where long-lived asteroids or co-orbital exoplanets can be found.

Equilibrium Solutions of the Logarithmic Hamiltonian Leapfrog for the N-body Problem

Abstract We prove that a second-order logarithmic Hamiltonian leapfrog for the classical general N-body problem (CGNBP) designed by Mikkola and Tanikawa and some higher-order logarithmic Hamiltonian methods based on symmetric multicompositions of the logarithmic algorithm exactly reproduce the orbits of elliptic relative equilibrium solutions in the original CGNBP. These methods are explicit symplectic methods. Before this proof, only some implicit discrete-time CGNBPs proposed by Minesaki had been analytically shown to trace the orbits of elliptic relative equilibrium solutions. The proof is therefore the first existence proof for explicit symplectic methods. Such logarithmic Hamiltonian methods with a variable time step can also precisely retain periodic orbits in the classical general three-body problem, which generic numerical methods with a constant time step cannot do.

Illustrating chaos: a schematic discretization of the general three-body problem in Newtonian gravity

We present a formalism for constructing schematic diagrams to depict chaotic three-body interactions in Newtonian gravity. This is done by decomposing each interaction in to a series of discrete transformations in energy- and angular momentum-space. Each time a transformation is applied, the system changes state as the particles re-distribute their energy and angular momenta. These diagrams have the virtue of containing all of the quantitative information needed to fully characterize most bound or unbound interactions through time and space, including the total duration of the interaction, the initial and final stable states in addition to every intervening temporary meta-stable state. As shown via an illustrative example for the bound case, prolonged excursions of one of the particles, which by far dominates the computational cost of the simulations, are reduced to a single discrete transformation in energy- and angular momentum-space, thereby potentially mitigating any computational expense. We further generalize our formalism to sequences of (unbound) three-body interactions, as occur in dense stellar environments during binary hardening. Finally, we provide a method for dynamically evolving entire populations of binaries via three-body scattering interactions, using a purely analytic formalism. In principle, the techniques presented here are adaptable to other three-body problems that conserve energy and angular momentum.

The rectilinear three-body problem as a basis for studying highly eccentric systems

The rectilinear elliptic restricted Three Body Problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity $e'=1$, but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter $\mu=0.5$ (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke's computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to $\mu$ and $e'<1$. Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.

High-velocity stars as a result of encounters between stars and massive binary black holes in galactic nuclei

Numerical simulations of the motions of stars in the gravitational fields of binary black holes with various component mass ratios have been carried out. Two models are considered: (1) the two-body problem with two fixed centers; (2) the general three-body problem. The first model is applicable only over short times Δt ≫ T, where T is the period of the binary system. The second model is applicable at all times except for during close encounters of stars with one of the binary components, r ≤ 0.00002 pc, where r is the distance from the star to the nearer black hole. In very close passages, relativistic corrections must be taken into account. Estimates of the probability of formation of high-velocity stars as a result of such interactions are obtained. It is shown that this mechanism is not suitable for the nucleus of our Galaxy due to the probable absence of a second massive black hole in the central region of the Galaxy.

Periodic orbits in the free-fall three-body problem

Periodic solutions of the general free-fall three-body problem are investigated for the case of equal masses. The initial conditions are chosen on a Hill surface in form space. The use of the form space reduces the dimension of the problem and makes it possible to represent the region of possible initial conditions on the Hill surface, together with a color scale. The regions obtained can be used to improve the precision of the initial conditions for the periodic orbits in the free-fall three-body problem.

Orbital stability of coplanar two-planet exosystems with high eccentricities

The long-term stability of the evolution of two-planet systems is considered by using the general three body problem (GTBP). Our study is focused on the stability of systems with adjacent orbits when at least one of them is highly eccentric. In these cases, in order for close encounters, which destabilize the planetary systems, to be avoided, phase protection mechanisms should be considered. Additionally, since the GTBP is a non-integrable system, chaos may also cause the destabilization of the system after a long time interval. By computing dynamical maps, based on Fast Lyapunov Indicator, we reveal regions in phase space with stable orbits even for very high eccentricities (e>0.5). Such regions are present in mean motion resonances (MMR). We can determine the position of the exact MMR through the computation of families of periodic orbits in a rotating frame. Elliptic periodic orbits are associated with the presence of apsidal corotation resonances (ACR). When such solutions are stable, they are associated with neighbouring domains of initial conditions that provide long-term stability. We apply our methodology so that the evolution of planetary systems of highly eccentric orbits is assigned to the existence of such stable domains. Particularly, we study the orbital evolution of the extrasolar systems HD 82943, HD 3651, HD 7449, HD 89744 and HD 102272 and discuss the consistency between the orbital elements provided by the observations and the dynamical stability.

Corr3p_tr: A particle approach for the general three-body problem

This work presents a convenient way to solve the non-relativistic Schrödinger equation numerically for a general three-particle system including full correlation and mass polarization. Both Coulombic and non-Coulombic interactions can be studied. The eigensolver is based on a second order dynamical system treatment (particle method). The Hamiltonian matrix never needs to be realized. The wavefunction evolves towards the steady state solution for which the Schrödinger equation is fulfilled. Subsequent Richardson extrapolations for several meshes are then made symbolically in matlab to obtain the continuum solution. The computer C code is tested under Linux 64 bit and both double and extended precision versions are provided. Test runs are exemplified and, when possible, compared with corresponding values in the literature. The computer code is small and self contained making it unusually simple to compile and run on any system. Both serial and parallel computer runs are straight forward. Program summaryProgram title: corr3p_trCatalogue identifier: AEYR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYR_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 15025No. of bytes in distributed program, including test data, etc.: 156430Distribution format: tar.gzProgramming language: ANSI C.Computer: Linux 64bit PC.Operating system: Linux 64bit.RAM: 300 M bytesClassification: 2.7, 2.8, 2.9.Nature of problem: The Schrödinger equation for an arbitrary three-particle system is solved using finite differences and a fast particle method for the eigenvalue problem [20, 21, 23].Solution method: A fast eigensolver is applied (see Appendix). This solver works for both symmetrical and nonsymmetrical matrices (which opens up for more accurate nonsymmetrical finite difference expressions to be applied at the boundaries). The three-particle Schrödinger equation is transformed in two major steps. First step is to introduce the function Q(r1,r2,μ)=r1r2(1−μ2)φ(r1,r2,μ), where μ=cos(012). The cusps (r1=r2,μ=1) are then transformed into boundary conditions. The derivatives of Q are then continuous in the whole computational space and thus the finite difference expressions are well defined. Three-particle coalescence (r1=r2=0,μ) is treated in the same way. The second step is to replace Q(r1,r2,μ) with (2x1x2)−1Q(x1,x2,μ). The space (x1,x2,μ) is much more appropriate for a finite difference approach since the square roots x1=r1,x2=r2 allow the boundaries to be much further out. The non-linearity of the x-grid also leads to a finer description near the nucleus and a coarser one further out thus resulting in a saving of grid points. Also, in contrast to the usual variable r12, we have instead used μ which is an independent variable. This simplifies the mathematics and numerical treatments. Several different grids can naturally run completely independent of each other thus making parallel computations trivial. From several grid results the physical property of interest is extrapolated to continuum space. The extrapolations are made in a matlab m-script where all computations can be made symbolically so the loss of decimal figures are minimized during this process. The computer code, including correlation effects and mass polarization, is highly optimized and deals with either triangular or quadratic domains in (x1,x2).Restrictions: The amount of CPU time may become unreasonable for states needing boundary conditions very far beyond the origin. Also if the condition number of the corresponding Hamiltonian matrix is very high, the number of iterations will grow. The use of double precision computations also puts a limit on the accuracy of extrapolated results to about 6–7 decimal figures.Unusual features: The numerical solver is based on a particle method presented in [20, 21, 23]. In the Appendix we provide specific details of dealing with eigenvalue problems. The program uses a 64 bit environment (Linux 64bit). Parallel runs can be made conveniently through a simple bash script.Additional comments: The discretized wavefunction is complete on every given grid. New interactions can therefore conveniently be added to the Hamiltonian without the need to seek for an appropriate basis set.Running time: Given a modern CPU such as Intel core i5 and that the outer boundary conditions of r1 and r2 is limited to, say 16 atomic units, the total CPU time of totally 10 grids of a serial run is typically limited to a few minutes. One can then expect about 6–7 correct figures in the extrapolated eigenvalue. A single grid of say h1=h2=h3=1/16 converges in less than 1 s (with an error in the eigenvalue of about 1 percent). Parallel runs are possible and can further minimize CPU times for more demanding tasks.

Fine structure of the region of initial conditions for close to periodic orbits in the general three-body problem

The region of initial conditions for close to periodic orbits is studied in the general three-body problem with components of equal mass and zero angular momentum. A method proposed earlier, based on minimization of a functional equal to the sum of the squares of the differences between the initial and current coordinates and velocities of the bodies, is used to search for such orbits. The search was conducted among orbits with periods T ≤ 2 000τ, where τ is the mean time for a component to cross the triple system. Elongated structures are found in the region of initial conditions, each of which corresponds to a certain periodic orbit. The detected structures seem to be conentrated along characteristic curves corresponding to the exact periodic orbits. A boundary zone of the initial conditions has been discovered, to the left and right of which orbits arising from the Schubart orbit and S orbit lie. Close to periodic orbits in the boundary zone possess the properties of both types of orbits. As a rule, these have periods of ~102τ. Examples of trajectories of the bodies are presented. Dynamical and geometrical properties of the studied orbits are described.

AN EFFICIENT CONSERVATIVE INTEGRATOR WITH A CHAIN REGULARIZATION FOR THE FEW-BODY PROBLEM

We design an efficient orbital integration scheme for the general N-body problem that preserves all the conserved quantities except the angular momentum. This scheme is based on the chain concept and is regarded as an extension of a d'Alembert-type scheme for constrained Hamiltonian systems. It also coincides with the discrete-time general three-body problem for particle number N = 3. Although the proposed scheme is only second-order accurate, it can accurately reproduce some periodic orbits, which cannot be done with generic geometric numerical integrators.