Binary-black-hole encounters, gravitational bursts, and maximum final spin

The spin of the final black hole in the coalescence of nonspinning black holes is determined by the "residual" orbital angular momentum of the binary. This residual momentum consists of the orbital angular momentum that the binary is not able to shed in the process of merging. We study the angular momentum radiated, the spin of the final black hole, and the gravitational bursts in a sequence of equal mass encounters. The initial orbital configurations range from those producing an almost direct infall to others leading to numerous orbits before infall, with multiple bursts of radiation. Our sequence consists of orbits with fixed impact parameter. What varies is the initial linear momentum of the black holes. For this sequence, the final black hole of mass M_{h} gets a maximum spin parameter a/M_{h} approximately 0.823, with this maximum occurring for initial orbital angular momentum L/M_{h};{2} approximately 1.176.     

One-dimensional three-body problem via symbolic dynamics

Symbolic dynamics is applied to the one-dimensional three-body problem with equal masses. The sequence of binary collisions along an orbit is expressed as a symbol sequence of two symbols. Based on the time reversibility of the problem and numerical data, inadmissible (i.e., unrealizable) sequences of collisions are systematically found. A graph for the transitions among various regions in the Poincare section is constructed. This graph is used to find an infinite number of periodic sequences, which implies an infinity of periodic orbits other than those accompanying a simple periodic orbit called the Schubart orbit. Finally, under reasonable assumptions on inadmissible sequences, we prove that the set of admissible symbol sequences forms a Cantor set. (c) 2000 American Institute of Physics.     

Geodesic Structure of a Non-linear Magnetic Charged Black Hole Surrounded by Quintessence

This paper aims to investigate the geodesic motion in the spacetime of a non-linear magnetic charged black hole surrounded by quintessence. By varying the Lagrangian corresponding to the metric, the orbital motion equation has been obtained. The effects of the magnetic charge $Q$, positive normalization factor $C$, angular momentum $b$, and energy $E$ on time-like and null geodesic motion are discussed from three aspects: orbital stability, orbital types, and circular orbits. By comparing the effects of the above parameters $C$, $b$ on the effective potential, it is found that quintessence has an impact on the types and stability of orbits. In addition, for time-like orbital motion, when $3.443113\leq b\leq6.392\,578$ (for fixed $C=0.0002$, $M = 1$, $Q=0.7$), there are bound orbits, and within this range, the stable circular orbits exist, and the radii of the innermost and outermost stable circular orbit are $r=5.912\,654$ and $r=56.745\,933$, respectively. For null orbital motion, the orbital types have only unstable circular orbit which occur at $r=2.951\,072$ ($E^{2}=E_{2}^{2}=0.4$), absorb orbits and escape orbits, but no stable circular orbits, and bound orbits.     

Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three Body Problem. Part II

We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem (PCR3BP) corresponding to the Sun-Jupiter-Oterma system possesses an infinite number of symmetric periodic orbits and homoclinic orbits to the Lyapunov orbits. Moreover, we show the existence of symbolic dynamics on six symbols for PCR3BP and the possibility of resonance transitions of the comet. This extends earlier results by Wilczak and Zgliczynski [12]. Electronic Supplementary Material: Supplementary material is available in the online version of this article at An erratum to this article is available at .     

Periodic orbits and binary collisions in the classical three-body Coulomb problem

In the helium case of the classical three-body Coulomb problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. Focusing our attention on binary collisions with these tools, a sequence of periodic orbits are predicted and are actually found numerically. A family of periodic orbits found has regularity in their actions. For this family of periodic orbits, it is shown that thanks to its regularity, a partial summation of the Gutzwiller trace formula with a daring approximation gives a Rydberg series of energy levels.     

Binary neutron stars: Equilibrium models beyond spatial conformal flatness

Equilibria of binary neutron stars in close circular orbits are computed numerically in a waveless formulation: the full Einstein-relativistic-Euler system is solved on an initial hypersurface to obtain an asymptotically flat form of the 4-metric and an extrinsic curvature whose time derivative vanishes in a comoving frame. Two independent numerical codes are developed, and solution sequences that model inspiraling binary neutron stars during the final several orbits are successfully computed. The binding energy of the system near its final orbit deviates from earlier results of third post-Newtonian and of spatially conformally flat calculations. The new solutions may serve as initial data for merger simulations and as members of quasiequilibrium sequences to generate gravitational-wave templates, and may improve estimates of the gravitational-wave cutoff frequency set by the last inspiral orbit.     

An ergodic algorithm for long-term coverage of elliptical orbits

This paper deals with the coverage analysis problem of elliptical orbits. An algorithm based on ergodic theory, for long-term coverage of elliptical orbits, is proposed. The differential form of the invariant measure is constructed via the perturbation on mean orbital elements resulted from the $J_{2}$ term of non-spherical shape of the earth. A rigorous proof for this is then given. Different from the case of circular orbits, here the flow and its space of the dynamical system are defined on a physical space, and the real-value function is defined as the characteristic function on station mask. Therefore, the long-term coverage is reduced to a double integral via Birkhoff--Khinchin theorem. The numerical implementation indicates that the ergodic algorithm developed is available for a wide range of eccentricities.     

An ergodic algorithm for long-term coverage of elliptical orbits

This paper deals with the coverage analysis problem of elliptical orbits. An algorithm based on ergodic theory, for long-term coverage of elliptical orbits, is proposed. The differential form of the invariant measure is constructed via the perturbation on mean orbital elements resulted from the J2 term of non-spherical shape of the earth. A rigorous proof for this is then given. Different from the case of circular orbits, here the flow and its space of the dynamical system are defined on a physical space, and the real-value function is defined as the characteristic function on station mask. Therefore, the long-term coverage is reduced to a double integral via Birkhoff-Khinchin theorem. The numerical implementation indicates that the ergodic algorithm developed is available for a wide range of eccentricities.     

6j Symbols for the Modular Double,Quantum Hyperbolic Geometry,and Supersymmetric Gauge Theories

We revisit the definition of the 6j symbols from the modular double of ${\mathcal{U}_q(\mathfrak{sl}(2, \mathbb{R}))}$ , referred to as b-6j symbols. Our new results are (1) the identification of particularly natural normalization conditions, and (2) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. Our new integral representations finally indicate a possible interpretation of the b-6j symbols as partition functions of non-abelian three-dimensional ${\mathcal{N}=2}$ supersymmetric gauge theories.     

Perfect Simulation of a Coupling Achieving the $\bar{d}$-distance Between Ordered Pairs of Binary Chains of Infinite Order

We explicitly construct a stationary coupling attaining Ornstein’s [`(d)]\bar{d}-distance between ordered pairs of binary chains of infinite order. Our main tool is a representation of the transition probabilities of the coupled bivariate chain of infinite order as a countable mixture of Markov transition probabilities of increasing order. Under suitable conditions on the loss of memory of the chains, this representation implies that the coupled chain can be represented as a concatenation of i.i.d. sequences of bivariate finite random strings of symbols. The perfect simulation algorithm is based on the fact that we can identify the first regeneration point to the left of the origin almost surely.     

Numerical experiments of the planar equal-mass three-body problem: the effects of rotation

The planar three-body problem with angular momentum is numerically and systematically studied as a generalization of the free-fall problem (i.e., the three-body problem with zero initial velocities). The initial conditions in the configuration space exhaust all possible forms of a triangle, whereas the initial conditions in the momentum space are chosen so that position vectors and momentum vectors are orthogonal. Numerical results are organized according to the value of virial ratio k defined as the ratio of the total kinetic energy to the total potential energy. Final motions are mapped in the initial value space. Several interesting features are found. Among others, binary collision curves seem to spiral into the Lagrange point, and for large k, binary collision curves connect the Lagrange point and the Euler point. The existence of a lunar periodic orbit and a periodic orbit of petal-type is suggested. The number of escape orbits as a function of the escape time is analyzed for different k. The behavior of this number for different time and k shows most remarkably the effects of rotation of triple systems. The number of escape orbits increases exponentially for k相似文献   

Photodetachment cross-section of $$\hbox {H}^{-}\,\hbox {ion}$$ in a three-dimensional cubical microcavity

The photodetachment of negative ions inside a two-dimensional microcavity has been studied by many researchers. As to the photodetachment of negative ions in the three-dimensional microcavity, the research is relatively little. In this paper, we study the photodetachment cross-section of \(\hbox {H}^{-}\) ion inside a three-dimensional cubical microcavity for the first time. We have observed the classical dynamics of the photodetached electron inside the cubical microcavity and found out its closed orbits. Then we calculate the photodetachment cross-section of this system. It is shown that owing to the interference effects of the electron wave travelling along various closed orbits, oscillatory structures appear in the photodetachment cross-section. And the oscillatory structures depend on the laser polarization sensitively. Compared to the photodetachment of \(\hbox {H}^{-}\) ion inside a square microcavity, in photodetachment of \(\hbox {H}^{-}\) ion in cubical cavity the number of the closed orbits is increased and the oscillatory structure in the photodetachment cross-section becomes much more complex. Through our study, researchers can gain a deep understanding on the correspondence of the classical dynamics and the quantum mechanics. Our study may guide future experimental research in the field of the photodetachment electron dynamics inside a three-dimensional microcavity.     

Shuffle Relations for Regularised Integrals of Symbols

We prove shuffle relations which relate a product of regularised integrals of classical symbols to regularised nested iterated integrals:

The effects of non-Kolmogorov turbulence on the orbital angular momentum of a photon-beam propagation in a slant channel

We analyze the effects of non-Kolmogorov turbulence on the orbital angular momentum of a photon-beam propagation through atmosphere. The probability models of the orbital angular momentum crosstalk for single photons propagation in the channel with the non-Kolmogorov turbulence aberration have been established. It is found that the crosstalk among orbits increases as the orbital angular momentum quantum number of launch beam rises, the ground turbulence strength ${C_n^{2} \left( 0 \right)}$ enhances or the non-Kolmogorov parameter α of turbulence-channel increases. As non-Kolmogorov parameter α approaches 4, the crosstalk probabilities among neighbor orbits are approximately the same.     

Electroexcitation of Low-Lying Particle-Hole RPA States of 16O with WBP Interaction

The nuclear structure of ¹⁶O is studied in the framework of the particle-hole random phase approximation (ph RPA). The Hamiltonian is diagonalized within a model space with particle orbits {1d_5/2,1d_3/2 , and 2s_1/2} and the hole orbits {1p_3/2 and 1p_1/2} using Warburton and Brown interaction WBP. The ph RPA calculations are tested, by comparing the electron scattering form factors with the available experimental data. The results of electron scattering form factors and reduced transition strength for the states: 1^-, T=0 (7.116 MeV); 2^-, T=1 (12.968 MeV); 2^-, T=1 (20.412 MeV); and 3^-, T=0 (6.129 MeV) are interpreted in terms of the harmonic-oscillator (HO) wave functions of size parameter b. The occupation probabilities of the single particle and hole orbits are calculated. The spurious states are removed by adding the center of mass (CM) correction to the nuclear Hamiltonian. A comparison with the available experiments data is presented.     

The elliptical motion of stars in close binary systems with a conservative mass exchange

The problem of the motion of a star in a close binary system with a conservative mass exchange is considered. In contrast to the well-known Paczynski-Huang model, a new model is used that defines the motion of close binary systems in the case of an elliptical orbit. The reactive forces and the force of gravity of stars by the overflowing jet are accounted for in the elliptic motion of the star. The calculations of elliptical orbits of close binary stars show that the effect of the reactive force on the evolution of the star orbit can be different. The changes of the major axis and eccentricity of the orbit of the second star are defined. The results are applied to the BF Aurigae star system and are given in the form of figures.     

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

The three-body problem can be traced back to Newton in 1687,but it is still an open question today.Note that only a few periodic orbits of three-body systems were found in 300 years after Newton mentioned this famous problem.Although triple systems are common in astronomy,practically all observed periodic triple systems are hierarchical(similar to the Sun,Earth and Moon).It has traditionally been believed that non-hierarchical triple systems would be unstable and thus should disintegrate into a stable binary system and a single star,and consequently stable periodic orbits of non-hierarchical triple systems have been expected to be rather scarce.However,we report here one family of 135445 periodic orbits of non-hierarchical triple systems with unequal masses;13315 among them are stable.Compared with the narrow mass range(only 10-5)in which stable"Figure-eight"periodic orbits of three-body systems exist,our newly found stable periodic orbits have fairly large mass region.We find that many of these numerically found stable non-hierarchical periodic orbits have mass ratios close to those of hierarchical triple systems that have been measured with astronomical observations.This implies that these stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses quite possibly can be observed in practice.Our investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses.Note that our approach has general meaning:in a similar way,every known family of periodic orbits of three-body systems with two or three equal masses can be used as a starting point to generate thousands of new periodic orbits of triple systems with distinctly unequal masses.     

Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space–time

Gravitational radiation of binary systems can be studied by using the adiabatic approximation in General Relativity. In this approach a small astrophysical object follows a trajectory consisting of a chained series of bounded geodesics (orbits) in the outer region of a Kerr Black Hole, representing the space time created by a bigger object. In our paper, we study the entire class of orbits, both of constant radius (spherical orbits), as well as non-null eccentricity orbits, showing a number of properties on the physical parameters and trajectories. The main result is the determination of the geometrical locus of all the orbits in the space of physical parameters in Kerr space–time. This becomes a powerful tool to know if different orbits can be connected by a continuous change of their physical parameters. A discussion on the influence of different values of the angular momentum of the hole is given. Main results have been obtained by analytical methods.     

Global Structure of Quaternion Polynomial Differential Equations

In this paper we mainly study the global structure of the quaternion Bernoulli equations \({\dot q=aq+bq^n}\) for \({q\in {\mathbb{H}}}\), the quaternion field and also some other form of cubic quaternion differential equations. By using the Liouvillian theorem of integrability and the topological characterization of 2–dimensional torus: orientable compact connected surface of genus one, we prove that the quaternion Bernoulli equations may have invariant tori, which possesses a full Lebesgue measure subset of \({{\mathbb{H}}}\). Moreover, if n = 2 all the invariant tori are full of periodic orbits; if n = 3 there are infinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits.