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

A quantitative technique for studying the localization of heavy-ion reactions in both coordinate ($r$) and angular momentum ($L$) space has been developed. A comparison of the degree of localization in elastic, inelastic, and one-particle transfer reactions is presented. For elastic scattering, the localization is studied as a function of the angular range of the data. A fully quantum mechanical relationship between $L$ and $r$ is demonstrated.[NUCLEAR REACTIONS Localization of heavy-ion reactions in coordinate and angular momentum space; elastic, inelastic, and one neutron transfer for $^{13}\mathrm{C}$ + $^{60}\mathrm{Ni}$ at ${E}_{\mathrm{lab}}=60.8$ MeV; elastic scattering of $^{16}\mathrm{O}$ + $^{28}\mathrm{Si}$ at ${E}_{\mathrm{lab}}=81$ MeV; optical model and DWBA analyses.

In this paper, we continue our analysis of the chaotic four-body problem and our study of binary–binary interactions in star clusters. We present a general ansatz-based analytical treatment using statistical mechanics, where each outcome of the four-body problem is regarded as some variation of the three-body problem. For example, when two single stars are produced (the 2 + 1 + 1 outcome), each ejection event is modelled as its own three-body interaction by assuming that the ejections are well separated in time. This is a generalization of the approach adopted in Paper I, based on the density-of-states formalism. There are three possible outcomes for the four-body problem with negative total energies: 2 + 2, 2 + 1 + 1, and 3 + 1. For each outcome, we apply an ansatz-based approach to deriving analytical distribution functions that describe the properties of the products of chaotic four-body interactions involving point particles. To test our theoretical distributions, we perform a set of scattering simulations in the equal-mass point-particle limit using FEWBODY, where we vary the initial ratio of binary semimajor axes. We compare our final theoretical distributions to the simulations for each particular scenario, finding consistently good agreement between the two. The highlights of our results include that binary–binary scatterings act to systematically destroy binaries producing instead a binary and two ejected stars (when the initial binary semimajor axes are similar) or a stable triple (when the initial semimajor axes are very different). The 2 + 2 outcome produces the widest binaries, and the 2 + 1 + 1 outcome produces the most compact binaries.

In this paper, we study the chaotic four-body problem in Newtonian gravity. Assuming point particles and total encounter energies $\le$ 0, the problem has three possible outcomes. We describe each outcome as a series of discrete transformations in energy space, using the diagrams first presented in Leigh \& Geller (2012; see the Appendix). Furthermore, we develop a formalism for calculating probabilities for these outcomes to occur, expressed using the density of escape configurations per unit energy, and based on the Monaghan description originally developed for the three-body problem. We compare this analytic formalism to results from a series of binary-binary encounters with identical point particles, simulated using the \texttt{FEWBODY} code. Each of our three encounter outcomes produces a unique velocity distribution for the escaping star(s). Thus, these distributions can potentially be used to constrain the origins of dynamically-formed populations, via a direct comparison between the predicted and observed velocity distributions. Finally, we show that, for encounters that form stable triples, the simulated single star escape velocity distributions are the same as for the three-body problem. This is also the case for the other two encounter outcomes, but only at low virial ratios. This suggests that single and binary stars processed via single-binary and binary-binary encounters in dense star clusters should have a unique velocity distribution relative to the underlying Maxwellian distribution (provided the relaxation time is sufficiently long), which can be calculated analytically.

There are periodic solutions to the equal-mass three-body (and N-body) problem in Newtonian gravity. The figure-eight solution is one of them. In this paper, we discuss its solution in the first and second post-Newtonian approximations to general relativity. To do so we derive the canonical equations of motion in the ADM gauge from the three-body Hamiltonian. We then integrate those equations numerically, showing that quantities such as the energy, linear and angular momenta are conserved down to numerical error. We also study the scaling of the initial parameters with the physical size of the triple system. In this way we can assess when general relativistic results are important and we determine that this occurs for distances of the order of 100M, with M the total mass of the system. For distances much closer than those, presumably the system would completely collapse due to gravitational radiation. This sets up a natural cut-off to Newtonian N-body simulations. The method can also be used to dynamically provide initial parameters for subsequent full nonlinear numerical simulations.

Understanding the formation and evolution of stellar-mass binary black holes (BBHs) requires a thorough investigation of the key physical processes involved. While one pathway involves the isolated evolution of massive binary stars, affected by uncertain stages like core-collapse supernovae and common envelope evolution, an alternative channel is dynamical formation in dense stellar environments. Newtonian gravity has traditionally provided a robust and computationally efficient framework for modeling large-scale gravitational interactions. However, accurately capturing close encounters and black hole mergers necessitates the use of general relativity. This work focuses on assessing the applicability of post-Newtonian gravity in bridging these regimes, offering a physically insightful and computationally tractable approach to modeling BBH formation in the gravitational-wave era of astronomy.This article is part of the theme issue 'Newton, Principia, Newton Geneva Edition (17th-19th) and modern Newtonian mechanics: heritage, past & present'.

The recent advances in observational and computational techniques allow studying the formation and dynamical evolution of planetary systems at an unprecedented level. The formation and evolution of isolated planetary systems are challenging in itself and it is more complicated by the dense environments in which stars and planets are typically born. Here, we present an overview of the internal and external processes that govern the dynamical evolution of planetary systems, and we provide a brief overview of a selection of the computational tools that are presently available to carry out realistic simulations of planetary systems in dense stellar environments.

Binary black hole (BBH) mergers are critical sources of gravitational waves detected by observatories like LIGO and Virgo. The formation and evolution of these binaries, particularly in dense stellar environments such as globular clusters (GCs) and galactic nuclei, are crucial for understanding the observed BBH population. In this study, we develop a Monte Carlo N-body simulation model to simulate the dynamical evolution of BBH mergers in such environments. We incorporate key processes, including two-body relaxation, three-body encounters, exchange interactions, and gravitational wave emission. Our results indicate that globular clusters can produce BBH merger rates of approximately 20 mergers per Gyr per 105M☉, in agreement with current gravitational wave observations. The simulated mass distribution of BBHs peaks at around 20 M ☉, with a tail extending to 50 M ☉, and the spin orientations are predominantly isotropic. These findings highlight the significant role dense stellar environments play in the formation of BBHs. We also discuss the uncertainties in the model and suggest future work to refine the simulations and better constrain the contribution of dense stellar environments to the overall BBH merger rate.

By numerically solving the single-particle stationary Schrödinger equation and the Gross-Pitaevskii equation with mean-field interactions at zero temperature, the ground state properties of the rotating spin-orbital-angular-momentum coupled Bose-Einstein condensates in a harmonic trapping potential are investigated in this work. The results show that the rotation lifts the double degeneracy of the single-particle energy spectrum in the angular momentum space, and leads to the vortex state. The angular momentum of the vortex depends on the rotating frequency, the intensity of the laser beam, and the spin-orbital-angular-momentum coupling. In particular, if the rotating frequency is below a critical value, the angular momentum of the ground state vortex remains unaffected by the rotating frequency. When the rotating frequency exceeds the critical value, the angular momentum of the ground state vortex will increase with the rotating frequency increasing. By assuming that the system is confined in a ring trap, the expression of the single-particle energy spectrum in the angular momentum space can be obtained, which clarifies how the rotation frequency affects the angular momentum of the ground state. In the presence of atomic interactions, similar phenomena can also be observed in the mean-field ground state at zero temperature.

Eccentric mergers are a signature of the dynamical formation channel of binary black holes (BBHs) in dense stellar environments and hierarchical triple systems. Here, we investigate the formation of eccentric mergers via binary-single interactions by means of 2.5 × 105 direct N-body simulations. Our simulations include post-Newtonian terms up to the 2.5th order and model the typical environment of young (YSCs), globular (GCs), and nuclear star clusters (NSCs). Around 0.6% (1%) of our mergers in NSCs (GCs) have an eccentricity > 0.1 when the emitted gravitational wave frequency is 10 Hz in the source frame, while in YSCs this fraction rises to 1.6%. Approximately ∼63% of these mergers are produced by chaotic, resonant interactions where temporary binaries are continuously formed and destroyed, while ∼31% arise from an almost direct collision of two black holes (BHs). Lastly, ∼6% of these eccentric mergers occur in temporary hierarchical triples. We find that binaries undergoing a flyby generally develop smaller tilt angles with respect to exchanges. This result challenges the idea that perfectly isotropic spin orientations are produced by dynamics. The environment dramatically affects BH retention: 0%, 3.1%, and 19.9% of all the remnant BHs remain in YSCs, GCs, and NSCs, respectively. The fraction of massive BHs also depends on the host cluster properties, with pair-instability (60 ≤ MBH/M⊙ ≤ 100) and intermediate-mass (MBH ≥ 100 M⊙) BHs accounting for approximately ∼44% and 1.6% of the mergers in YSCs, ∼33% and 0.7% in GCs, and ∼28% and 0.4% in NSCs, respectively.

Consider a system of point masses in a spherical potential. In such systems objects execute planar orbits covering two-dimensional rings or annuli, represented by the angular-momentum vectors, which slowly reorient due to the persistent weak gravitational interaction between different rings. This process, called vector resonant relaxation, is much faster than other processes which change the size/shape of the rings. The interaction is strongest between objects with closely aligned angular-momentum vectors. In this paper, we show that nearly parallel angular-momentum vectors may form stable bound pairs in angular-momentum space. We examine the stability of such pairs against an external massive perturber, and determine the critical separation analogous to the Hill radius or tidal radius in the three-body problem, where the angular-momentum pairs are marginally disrupted, as a function of the perturber’s mass, the orbital inclination, and the radial distance. Angular-momentum pairs or multiples closer than the critical inclination will remain bound and evolve together in angular-momentum-direction space under any external influence, such as anisotropic density fluctuations, or massive perturbers. This study has applications in various astrophysical contexts, including galactic nuclei, in particular the Milky Way’s Galactic Centre, globular clusters, or planetary systems. In nuclear star clusters with a central supermassive black hole, we apply this criterion to the disc of young, massive stars, and show that clusters in angular-momentum space may be used to constrain the presence of intermediate-mass black holes or the mass of the nearby gaseous torus.

This paper uses Gaussian mixture techniques to dissect the Milky Way (MW) stellar halo in angular momentum space. Application to a catalogue of 5389 stars near the plane of the Sagittarius (Sgr) stream with full 6D phase-space coordinates supplied by Gaia EDR3 and SEGUE returns four independent dynamical components. The broadest and most populated corresponds to the smooth MW halo. The narrowest and faintest contains 40 stars of the Orphan stream. We find a component with little or no angular momentum likely associated with the Gaia-Sausage-Enceladus substructure. We also identify 925 stars and 7 globular clusters with probabilities $\gt 90{{\ \rm per\ cent}}$ to be members of the Sgr stream. Comparison against N-body models shows that some of these members trace the continuation of the leading/trailing tails in the Southern/Northern hemispheres. The new detections span ∼800° on the sky, thus wrapping the Galaxy twice.

Representation of permutation operators in quantum mechanics

Context. The local stellar halo of the Milky Way contains the debris from several past accretion events. Aims. Here we study in detail the structure and properties of nearby debris associated with the Helmi streams, which was originally identified as an overdensity in integrals of motion space. Methods. We use 6D phase-space information from Gaia EDR3 combined with spectroscopic surveys, and we analyse the orbits and frequencies of the stars in the streams using various Galactic potentials. We also explore how the Helmi streams constrain the flattening, q, of the Galactic dark matter halo. Results. We find that the streams are split into substructures in integrals of motion space, most notably into two clumps in angular momentum space. The clumps have consistent metallicity distributions and stellar populations, supporting a common progeny. In all the realistic Galactic potentials explored, the Helmi streams’ stars depict a diffuse distribution close to Ωz/ΩR ∼ 0.7. At the same time, the reason for the substructure in angular momentum space appears to be a Ωz : Ωϕ resonance close to 1:1. This resonance is exactly 1:1 in the case where the (density) flattening of the dark halo is q = 1.2. For this halo shape, the substructure in angular momenta is also long-lasting. Conclusions. Our findings suggest that the structure of the Galactic potential leaves a clear imprint on the properties of phase-mixed debris streams.

The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more than 300 years. In this paper, we present a review of the three-body problem in the context of both historical and modern developments. We describe the general and restricted (circular and elliptic) three-body problems, different analytical and numerical methods of finding solutions, methods for performing stability analysis and searching for periodic orbits and resonances. We apply the results to some interesting problems of celestial mechanics. We also provide a brief presentation of the general and restricted relativistic three-body problems, and discuss their astronomical applications.

Recent years, it has been required accurate and agile attitude control of satellites. For this purpose, the necessity of Control Moment Gyros (CMGs) has been increasing, which can generate much higher torque than Reaction Wheel which is used for conventional spacecraft actuator. CMGs have singularity problem that they cannot output desired torque when they try to output high torque. When CMGs close to singularity, it's dangerous that its gimbal moves very quickly and widely. It is necessary to avoid singularity. In this paper, we focus on singularity and CMGs' angular momentum and propose a singularity avoidance method by planning path in angular momentum space. Because output torque from CMGs depends on path of angular momentum, we plan the angular momentum path that avoids singularity and its length is short as possible by application of A* algorithm. It is shown from the simulations that the proposed method can realize singularity avoidance and low gimbal drive.