Narrow Results

The second (unphysical) critical charge in the three-body quantum Coulomb system of a nucleus of positive charge $Z$ and mass $m_p$, and two electrons, predicted by Stillinger has been calculated to be equal to $Z_B^{\infty }=0.904854$ and $Z_B^{m_p}=0.905138$ for infinite and finite (proton) mass $m_p$, respectively. It is shown that in both cases, the ground state energy $E(Z)$ (analytically continued beyond the first critical charge $Z_c$, for which the ionization energy vanishes, to $\mathrm{\text{Re}}Z$$<$$Z_c$) has a square-root branch point with exponent 3/2 at $Z=Z_B$ in the complex $Z$-plane. Based on analytic continuation, the second, excited, spin-singlet bound state of negative hydrogen ion H${}^{-}$ is predicted to be at −0.51554 a.u. (−0.51531 a.u. for the finite proton mass $m_p$). The first critical charge $Z_c$ is found accurately for a finite proton mass $m_p$ in the Lagrange mesh method, $Z_c^{m_p}=0.911069724655$.

Three-body cluster-model calculations are performed for the new types of big-bang nucleosynthesis (BBN) reactions that are calalyzed by a supersymmetric (SUSY) particle stau, a scalar partner of the tau lepton. If a stau has a lifetime ≳ 10³s, it would capture a light element previously synthesized in standard BBN and form a Coulombic bound state. The bound state, an exotic atom, is expected to induce various reactions, such as (αX^-) + d → ⁶Li + X^-, in which a negatively charged stau (denoted as X^-) works as a catalyzer. Recent literature papers have claimed that some of these stau-catalyzed reactions have significantly large cross sections so that inclusion of the reactions into the BBN network calculation can change drastically abundances of some elements, giving not only a solution to the ⁶Li-⁷Li problem (calculated underproduction of ⁶Li by ~ 1000 times and overproduction of ⁷Li+⁷Be by ~ 3 times) but also a constraint on the lifetime and the primordial abundance of the elementary particle stau. However, most of these literature calculations of the reaction cross sections were made assuming too naive models or approximations that are unsuitable for those complicated low-energy nuclear reactions. We use a few-body calculational method developed by the authors, and provides precise cross sections and rates of the stau-catalyzed BBN reactions for the use in the BBN network calculation.

We consider a restricted three-body problem, where two interacted particles are located in a two-dimensional (2D) plane and interact with the third one located in the parallel spatially separated plane. The system of such type can be formed in the semiconductor coupled quantum wells, where the electrons (holes) and direct excitons spatially separated in different parallel neighboring quantum wells are sufficiently close to interact and form negative X^- or positive X⁺ indirect trions. It is shown that at large interwell separations, when the interwell separation is much greater than the exciton Bohr radius, this problem can be solved analytically using the cluster approach. Analytical results for the energy spectrum and the wavefunctions of the spatially indirect trion are obtained, their dependencies on the interwell separations is analyzed and a conditional probability distribution is calculated. The formation of 2D Wigner crystal of trions at the low densities is predicted. It is shown that the critical density of the formation of the trion Wigner crystal is sufficiently greater than the critical density of the electron Wigner crystal in the same material.

We show how to compute families of periodic solutions of conservative systems with two-point boundary value problem continuation software. The computations include detection of bifurcations and corresponding branch switching. A simple example is used to illustrate the main idea. Thereafter we compute families of periodic solutions of the circular restricted 3-body problem. We also continue the figure-8 orbit recently discovered by Chenciner and Montgomery, and numerically computed by Simó, as the mass of one of the bodies is allowed to vary. In particular, we show how the invariances (phase-shift, scaling law, and x, y, z translations and rotations) can be dealt with. Our numerical results show, among other things, that there exists a continuous path of periodic solutions from the figure-8 orbit to a periodic solution of the restricted 3-body problem.

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L₁ between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L₁ and the other collinear libration points L₂, L₃ is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

During the winter 1679, R. Hooke challenged I. Newton to predict the dynamics of an object submitted to a constant radial force. This correspondence made a strong impact on I. Newton, who wrote four years later "De Motu", the real ancestor of "The Principia", published in 1687. R. Hooke's problem can be physically linked to the dynamics of a sphere sliding on an inverted cone due to gravitational effects. If the symmetry axis of the cone is parallel to the gravitational field, the ball executes stable precessions. Breaking this symmetry induces the appearance of chaotic motions. After having derived the equations related to the position of the sphere, we analyze its dynamics, and we perform an approximated Floquet analysis that is compared to our numerical results.

In the three-body problem, it is known that there exists a special set of periodic orbits: spatial isosceles periodic orbits. In each period, one body moves up and down along a straight line, and the other two bodies rotate around this line. In this work, we revisit this set of orbits by applying variational method. Two unexpected phenomena are discovered. First, this set is not always spatial. It actually bifurcates from the circular Euler (central configuration) orbit to the Broucke (collision) orbit. Second, one of the orbits in this set encounters an oscillating behavior. By running its initial condition, the orbit stays periodic for only a few periods before it becomes irregular. However, it moves close to another periodic shape in a while. Shortly it falls apart again and starts running close to a third periodic shape after a moment. This oscillation continues as t increases. Actually, up to t = 1.2 × 10⁵, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions.

This work studies periodic orbits as action minimizers in the spatial isosceles three-body problem with mass $M=[1,m,1]$. In each period, the body with mass $m$ moves up and down on a vertical line, while the other two bodies have the same mass $1$, and rotate about this vertical line symmetrically. For given $m>0$, such periodic orbits form a one-parameter set with a rotation angle $\theta$ as the parameter.

Two new phenomena are found for this set. First, for each $m>0$, this set of periodic orbits bifurcate from a circular Euler (central configuration) orbit to a Broucke (collision) orbit as $\theta$ increases from $0$ to $\pi$. There exists a critical rotation angle ${\theta }_0(m)$, where the orbit is a circular Euler orbit if $0<\theta \le {\theta }_0(m)$; a spatial orbit if ${\theta }_0(m)<\theta <\pi$; and a Broucke (collision) orbit if $\theta =\pi$. The exact formula of ${\theta }_0(m)$ is numerically proved to be ${\theta }_0(m)=\frac{\pi }{4}\sqrt{\frac{4m+1}{m+2}}$. Second, oscillating behaviors occur at rotation angle $\theta =\pi /2$ for all $m\in [0.1,3]$. Actually, the orbit with $\theta =\pi /2$ runs on its initial periodic shape for only a few periods. It breaks the first periodic shape and becomes irregular in a moment. However, it runs close to a different periodic shape after a while. In a short time, it falls apart from the second periodic shape and runs irregularly again. Such oscillation continues as time $t$ increases. Up to $t=1.2\times 10^5$, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions. Further study implies that, for each $m\in [0.1,3]$, the angle between any two consecutive periodic shapes is a constant. When $m>3$, similar oscillating behaviors are expected.

In this paper, we study the $n$-body problem in the Poincaré upper half-plane ${ℍ}_R^2$, where the radius $R$ of the Poincaré disk is fixed. We introduce a new potential to derive the condition for hyperbolic relative equilibria on ${ℍ}_R^2$. We analyze the relative equilibrium of positive masses moving along geodesics under the $\text{SL}(2,ℝ)$ group. This result is utilized to establish the existence of relative equilibria for the $n$-body problem on ${ℍ}_R^2$ for $n=2$ and $n=3$. We revisit previously known results and uncover new qualitative findings on relative equilibria that are not evident in an extrinsic context. Additionally, we provide a simple expression for the center of mass of a system of point particles on a two-dimensional surface with negative constant Gaussian curvature.

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.

In the framework of general relativity, we discuss choreographic solutions for the three-body problem, where a solution is called choreographic if every massive particles move periodically in a single closed orbit. In general relativity, the periastron shift prohibits a binary system from orbiting in a single closed curve. Remarkably, a “figure-eight” solution is shown to be choreographic even at the PN approximation by carefully examining initial conditions. Next, gravitational waves for two- and three-body gravitating systems are discussed as an inverse problem. It is shown that quadrupole waveforms cannot distinguish these sources at particular configurations, especially through extending the definition of the chirp mass to such a three-body system. Finally, we present a conjecture on N particles for classification of sources with multipolar waveforms.

We reexamine the post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem. For three finite masses, it is found that a triangular configuration satisfies the post-Newtonian equation of motion in general relativity, if and only if it has the relativistic corrections to each side length. For the same masses and angular velocity, the post-Newtonian triangular configuration is always smaller than the Newtonian one.