Gravitational Three-body Problem Research Articles

Isles of regularity in a sea of chaos amid the gravitational three-body problem

Context. The three-body problem (3BP) poses a longstanding challenge in physics and celestial mechanics. Despite the impossibility of obtaining general analytical solutions, statistical theories have been developed based on the ergodic principle. This assumption is justified by chaos, which is expected to fully mix the accessible phase space of the 3BP. Aims. This study probes the presence of regular (i.e. non-chaotic) trajectories within the 3BP and assesses their impact on statistical escape theories. Methods. Using three-body simulations performed with the accurate, regularized code TSUNAMI, we established criteria for identifying regular trajectories and analysed their impact on statistical outcomes. Results. Our analysis reveals that regular trajectories occupy a significant fraction of the phase space, ranging from 28% to 84% depending on the initial setup, and their outcomes defy the predictions of statistical escape theories. The coexistence of regular and chaotic regions at all scales is characterized by a multi-fractal behaviour. Integration errors manifest as numerical chaos, artificially enhancing the mixing of the phase space and affecting the reliability of individual simulations, yet preserving the statistical correctness of an ensemble of realizations. Conclusions. Our findings underscore the challenges in applying statistical escape theories to astrophysical problems, as they may bias results by excluding the outcome of regular trajectories. This is particularly important in the context of formation scenarios of gravitational wave mergers, where biased estimates of binary eccentricity can significantly impact estimates of coalescence efficiency and detectable eccentricity.

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.

Stability analysis of planetary systems via second-order Rényi entropy

ABSTRACT The long-term dynamical evolution is a crucial point in recent planetary research. Although the amount of observational data are continuously growing and the precision allows us to obtain accurate planetary orbits, the canonical stability analysis still requires N-body simulations and phase space trajectory investigations. We propose a method for stability analysis of planetary motion based on the generalized Rényi entropy obtained from a scalar measurement. The radial velocity data of the central body in the gravitational three-body problem are used as the basis of a phase space reconstruction procedure. Then, Poincaré’s recurrence theorem contributes to finding a natural partitioning in the reconstructed phase space to obtain the Rényi entropy. It turns out that the entropy-based stability analysis is in good agreement with other chaos detection methods, and it requires only a few tens of thousands of orbital period integration time.

A Canonical Transformation to Eliminate Resonant Perturbations. I.

Abstract We study dynamical systems that admit action-angle variables at leading order, which are subject to nearly resonant perturbations. If the frequencies characterizing the unperturbed system are not in resonance, the long-term dynamical evolution may be integrated by orbit-averaging over the high-frequency angles, thereby evolving the orbit-averaged effect of the perturbations. It is well known that such integrators may be constructed via a canonical transformation, which eliminates the high-frequency variables from the orbit-averaged quantities. An example of this algorithm in celestial mechanics is the von Zeipel transformation. However, if the perturbations are inside or close to a resonance, i.e., the frequencies of the unperturbed system are commensurate; these canonical transformations are subject to divergences. We introduce a canonical transformation that eliminates the high-frequency phase variables in the Hamiltonian without encountering divergences. This leads to a well-behaved symplectic integrator. We demonstrate the algorithm through two examples: a resonantly perturbed harmonic oscillator and the gravitational three-body problem in mean motion resonance.

Flux-based statistical prediction of three-body outcomes

The gravitational three-body problem is a rich open problem, dating back to Newton. It serves as a prototypical example of a chaotic system and has numerous applications in astrophysics. Generically, the motion is non-integrable and susceptible to disintegration, and for negative total energy the decay outcome is a free body flying apart from a binary. Since Poincar\'e, the problem is known to be chaotic and is believed to lack a general deterministic solution. Instead, decades ago a statistical solution was marked as a goal. Yet, despite considerable progress, all extant approaches display two flaws. First, probability was equated with phase space volume, thereby ignoring the fact that significant regions of phase space describe regular motion, including post-decay motion. Secondly and relatedly, an adjustable parameter, the strong interaction region, which is a sort of cutoff, was a central ingredient of the theory. This paper introduces remedies and presents for the first time a statistical prediction of decay rates, in addition to outcomes. Based on an analogy with a particle moving within a leaky container, the statistical distribution is presented in an exactly factorized form. One factor is the flux of phase-space volume, rather than the volume itself, and it is given in a cutoff-independent closed-form. The other factors are the chaotic absorptivity and the regularized phase space volume. The situation is analogous to Kirchhoff's law of thermal radiation, also known as greybody radiation. In addition, an equation system for the time evolution of the statistical distribution is introduced; it describes the decay rate statistics while accounting for sub-escape excursions. Early numerical tests indicate a leap in accuracy.

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.

A Series Solution Approach to the Circular Restricted Gravitational Three-Body Dynamical Problem

The present manuscript examines the circular restricted gravitational three-body problem (CRGTBP) by the introduction of a new approach through the power series method. In addition, certain computational algorithms with the aid of Mathematica software are specifically designed for the problem. The algorithms or rather mathematical modules are established to determine the velocity and position of the third body’s motion. In fact, the modules led to accurate results and thus proved the new approach to be efficient.

Relation between solutions of the Schrödinger equation with transitioning resonance solutions of the gravitational three-body problem

It is shown that a class of approximate resonance solutions in the three-body problem under the Newtonian gravitational force are equivalent to quantized solutions of a modified Schrödinger equation for a wide range of masses that transition between energy states. In the macroscopic scale, the resonance solutions are shown to transition from one resonance type to another through weak capture at one of the bodies, while in the Schrödinger equation, one obtains quantized wave solutions transitioning between different energies. The resonance transition dynamics provides a classical model of a particle moving between different energy states in the Schrödinger equation. This methodology provides a connection between celestial and quantum mechanics.

Non-integrability of the dumbbell and point mass problem

This paper discusses a constrained gravitational three-body problem with two of the point masses separated by a massless inflexible rod to form a dumbbell. The non-integrability of this system is proven using differential Galois theory.

Non-integrability of the three-body problem

We consider the planar problem of three bodies which attract mutually with the force proportional to a certain negative integer power of the distance between the bodies. We show that such generalisation of the gravitational three-body problem is not integrable in the Liouville sense.

Hyperbolic Geometry for the Binary Collision Angles of the Three-Body Problem in the Plane

When the gravitational three-body problem in the plane is studied in the Pina and Jimenez system of coordinates, the shape sphere is simply related to the coordinates. This shape sphere is considered as a four 2-sphere, where contrary to other authors, the poles of the sphere occur for two equal inertia moments, not for the Lagrange points. In this sphere the central configurations correspond to fixed points of the sphere which are functions of the three masses. The binary collisions are also represented as points in this sphere. The hyperbolic geometry of the collision angles is the link between the instantaneous moments of inertia and the distances between particles.

Physical and dynamical parameters of the multiple system HD 222326

We present the results of our study of the physical and dynamical parameters of the multiple system HD 222326. A new method for determining the individual radial velocities of components in wide binary and multiple systems in the case of small radial-velocity differences (δV r ≤ the FWHMfor the line profiles) is suggested and tested for both model systems and the binary HD 10009. This testing yielded the component radial velocities V r 1,2 for HD 10009, enabling us to derive the center-of mass velocity, V γ, for the first time. We determined the radial velocities of the components of HD 222326 from high-resolution spectra, and refined the orbital parameters of the subsystems using speckle-interferometric observations. A combined spectroscopic and speckle interferometric analysis enabled us to find the positions of the components in the spectral type—luminosity diagram and to estimate their masses. It is likely that the components are all in various evolutionary stages after leaving the main sequence. We analyzed the dynamical evolution of the system using numerical modeling in the gravitational three-body problem and the known stability criteria for triple systems. The system is probably stable on time scales of at least 106 years. The presence of a fourth component in the system is also suggested.

Geometric integrators for multiple time-scale simulation

In this paper, we review and extend recent research on averaging integrators for multiple time-scale simulation such as are needed for physical N-body problems including molecular dynamics, materials modelling and celestial mechanics. A number of methods have been proposed for direct numerical integration of multiscale problems with special structure, such as the mollified impulse method (Garcia-Archilla, Sanz-Serna and Skeel 1999 SIAM J. Sci. Comput. 20 930?63) and the reversible averaging method (Leimkuhler and Reich 2001 J. Comput. Phys. 171 95?114). Features of problems of interest, such as thermostatted coarse-grained molecular dynamics, require extension of the standard framework. At the same time, in some applications the computation of averages plays a crucial role, but the available methods have deficiencies in this regard. We demonstrate that a new approach based on the introduction of shadow variables, which mirror physical variables, has promised for broadening the usefulness of multiscale methods and enhancing accuracy of or simplifying computation of averages. The shadow variables must be computed from an auxiliary equation. While a geometric integrator in the extended space is possible, in practice we observe enhanced long-term energy behaviour only through use of a variant of the method which controls drift of the shadow variables using dissipation and sacrifices the formal geometric properties such as time-reversibility and volume preservation in the enlarged phase space, stabilizing the corresponding properties in the physical variables. The method is applied to a gravitational three-body problem as well as a partially thermostatted model problem for a dilute gas of diatomic molecules.

Coorbital Dynamics at Large Eccentricity and Inclination

We show that coorbital dynamics at large eccentricity and inclination exhibit hitherto unknown types of stable motion in the gravitational three-body problem. In the solar system, the secular stability is observed in the evolution of specific near-Earth asteroids. In particular, the slow diffusion of such objects through the Earth's coorbital region which leads to temporary capture suggests the existence of undiscovered retrograde moons of the Earth. In the Coulomb problem, the Kozai resonance is a primary candidate for stable low angular momentum states in two-electron atoms.

The restricted gravitational three-body problem trajectories associated with Lagrange fixed points

Studies of the restricted three-body problem can help in understanding the dynamics of three-body interactions in the solar system. In the rotating coordinate system based on the two principal bodies there are five fixed Lagrange points. The equations of motion of the third body near a Lagrange point may be linearized, showing that under certain conditions there can be librational motion about the Lagrange points. Some examples of numerical studies of trajectories associated with Lagrange points are shown in inertial and in rotating coordinates, and are discussed. The examples include librations, horseshoe orbits and chaotic orbits. The restricted three-body problem, with its four degrees of freedom, provides considerable opportunities for project work in computational physics.} (Serbian). Rad na analizi problema tri tela u ogranicenom slucaju moze pomoci boljem razumevanju dinamike tri tela u suncevom sistemu. U koordinatnom sistemu koji rotira, vezanim za dva primarna tela, postoje pet stacionarnih Lagrangevih tacaka. Jednacine kretanja treceg tela u blizini Lagranzeve tacke mogu biti linearizovane, I pod izvesnim uslovima pokazuju da moze postojati libraciono kretanje oko Lagranzevih tacaka. Prikazani su primeri numerickih resenja putanja oko Lagranzevih tacaka u inercijalnim I rotirajucim koordinatama. Primeri ukljucuju libracije, potkovicne orbite I haoticne orbite. Ograniceni problem tri tela, sa cetiri stepena slobode pruza znacajne mogucnosti za studentske projekte iz numericke fizike.

Nonperiodic variations in astrophysical systems: Investigating frequency evolution

view Abstract Citations (11) References (25) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Nonperiodic Variations in Astrophysical Systems: Investigating Frequency Evolution Boyd, Patricia T. ; Carter, Patricia H. ; Gilmore, Robert ; Dolan, Joseph F. Abstract We present a method related to the wavelet transform, the Gabor transform, for investigating astronomical time series containing non-consistent frequencies. Instances in which such data sets may arise include variable star light curves, numerical studies of the gravitational three-body problem, X-ray binaries, and signals from more exotic objects such as planets around pulsars and mass infall from accretion structures onto compact objects. As an illustration of its power, we apply the technique to a numerical data set of a gravitational three-body interaction and to photometry of the rapidly oscillating peculiar A star HD 60435. In the three-body example, the method provides an insightful shorthand that allows for the determination of episodes where the system behaves as two nearly Keplerian orbits. For HD 60435, the power in the main frequency exhibits unusual evolution over the duration of the observation. Publication: The Astrophysical Journal Pub Date: June 1995 DOI: 10.1086/175746 Bibcode: 1995ApJ...445..861B Keywords: Celestial Mechanics; Frequency Distribution; Peculiar Stars; Periodic Variations; Stellar Oscillations; Time Series Analysis; Wavelet Analysis; Gravitational Effects; Light Curve; Time Dependence; Transformations (Mathematics); X Ray Binaries; Astrophysics; CELESTIAL MECHANICS; STELLAR DYNAMICS; METHODS: ANALYTICAL; STARS: OSCILLATIONS; STARS: INDIVIDUAL HENRY DRAPER NUMBER: HD 60535; STARS: PECULIAR full text sources ADS | data products SIMBAD (2)

The geometry of the roche coordinates and zero-velocity curves in the photogravitational three-body problem

The aim of the paper is to study the geometry of the Roche curvilinear coordinates (ξ, η, ζ) in the photogravitational circular restricted three-body problem, with varying radiation pressure, and special attention is given to the geometry of zero-velocity curves specified by theξ coordinate. The radiation pressure exerted by the primary bodies on the infinitesimal third body is considered the same (q 1 =q 2), and the primaries are taken to have equal masses (m 1 =m 2). The full range of values of the common radiation factor is explored, from the valueq 1 =q 2 = 1 (the gravitational three-body problem) down toq 1 =q 2 ≅ 0. It is found that radiation has a strong influence on the geometry of the Roche coordinates and the zero-velocity curves.

Chaotic scattering in the gravitational three-body problem.

We summarize some results of an ongoing study of the chaotic scattering interaction between a bound pair of stars (a binary) and an incoming field star. The stars are modeled as point masses and their equations of motion are numerically integrated for a large number of initial conditions. The global features of the resulting initial-value space maps are presented, and their evolution as a function of system parameters is discussed. We find that the maps contain regular regions separated by rivers of chaotic behavior. The probability of escape within the chaotic regions is discussed, and a straightforward explanation of the scaling present in these regions is reviewed. We investigate a statistical quantity of interest, namely the cross section for temporarily bound interactions, as a function of the third star's incoming velocity and mass. Finally, a new way of considering long-lived trajectories is presented, allowing long data sets to be qualitatively analyzed at a glance.

Chaos in the one-dimensional gravitational three-body problem.

We have investigated the appearance of chaos in the one-dimensional Newtonian gravitational three-body system (three masses on a line with -1/r pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincare sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18 000 full orbits (with initial states on a 100x180 lattice on the Poincare section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincare maps for 10x18 starting points. Our results show that the Poincare section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: (1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of 'scallops' bordering the E=0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. (2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincare section. For both (1) and (2) the initial and final states consist of a binary + single particle. (3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.

On a Result of Bruns

AbstractBruns' Theorem states that the classical integrals of the gravitational three-body problem generate all algebraic integrals. We show that the first step in his proof, together with Ziglin's non-integrability criterion for complex systems, can be used to prove the non-existence of energy independent algebraic integrals in certain real analytic systems. We also show that this aspect of Bruns' argument is purely algebraic: We offer a proof based on elementary differential algebraic methods.