Copenhagen Problem Research Articles

Determination of the doubly symmetric periodic orbits in the restricted three-body problem and Hill’s lunar problem

We review some recent progress on the research of the periodic orbits of the N-body problem, and numerically study the spatial doubly symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type SDSPs in the circular restricted three-body problem are computed, as well as the Hill-type SDSPs in Hill’s lunar problem. Double symmetries are exploited so that the SDSPs can be computed efficiently. The monodromy matrix can be calculated by the information of one fourth period. The periodicity conditions are solved by Broyden’s method with a line-search, and some numerical examples show that the scheme is very efficient. For a fixed period ratio and a given acute angle, there exist sixteen cases of initial values. For the restricted three-body problem, the cases of “Copenhagen problem” and the Sun–Jupiter–asteroid model are considered. New SDSPs are also numerically found in Hill’s lunar problem. Though the period ratio should be small theoretically, some new periodic orbits are found when the ratio is not too small, and the linear stability of the searched SDSPs is numerically determined.

Analysis of Copenhagen problem with a repulsive quasi‐homogeneous Manev‐type potential within the frame of variable mass

AbstractThe present problem is devoted to the special case of restricted problem of three bodies, always referred as the Copenhagen problem when we consider the primaries of equal masses. In this manuscript, we have considered the third body of infinitesimal mass as variable and moreover, instead of taking only Newtonian potential and forces, a quasi‐homogeneous potential produced by the main primaries has also been taken into account. This implies that, in order to approximate the phenomena such as the radiation pressure or the primaries' nonsphericity, we insert an inverse cube corrective term to the gravitational law of inverse square. In this model, the impact of the parameters, arise as a result of variable mass on the number and locations of the in‐plane and out‐of‐plane equilibrium points, is discussed. Moreover, their stability is also analyzed and the impact of these parameters on the regions, where motion of the third particle is possible, is unveiled. The basins of convergence associated with these equilibrium points are also illustrated by the Newton–Raphson iterative scheme of two or more variables.

Orbital analysis in the planar circular Copenhagen problem using polar coordinates

We examine the orbital dynamics of the planar Copenhagen problem, by classifying sets of starting conditions of orbits, expressed in polar coordinates. Specifically, we examine how the Jacobi constant influences several aspects of the overall dynamics of the test particle, such as its final state, as well as the time of escape/collision. By using modern diagrams with color codes, we manage to present the different types of basins. It is shown that both the character of the orbits and the fractal degree of the system are highly dependent on the Jacobi constant.

Orbit classification in the Copenhagen problem with oblate primaries

AbstractThis article aims to numerically investigate the orbital dynamics of the planar circular‐restricted three‐body problem, where the two primary bodies have equal masses and an oblate spheroid shape. By numerically integrating several large sets of initial conditions of both retrograde and prograde orbits, we classify them into three main categories: (a) bounded (regular or chaotic), (b) escaping, and (c) collision orbits. We reveal the motion of the test particle by presenting color‐coded diagrams, where the initial conditions are mapped to the orbit type and studied as a function of the total orbital energy (or equivalently of the Jacobi constant) as well as of the oblateness coefficient. Our numerical results are compared with related previous ones, corresponding to the classical version of the planar‐restricted three‐body problem. The presented outcomes have practical physical applications as they can be used for modeling the motion of a massless test particle in the gravitational field of a binary system of two oblate exoplanets.

Orbit classification in a pseudo-Newtonian Copenhagen problem with Schwarzschild-like primaries

We examine the orbital dynamics of the planar pseudo-Newtonian Copenhagen problem, in the case of a binary system of Schwarzschild-like primaries, such as super-massive black holes. In particular, we investigate how the Jacobi constant (which is directly connected with the energy of the orbits) influences several aspects of the orbital dynamics, such as the final state of the orbits. We also determine how the relativistic effects (i.e., the Schwarzschild radius) affect the character of the orbits, by comparing our results with the classical Newtonian problem. Basin diagrams are deployed for presenting all the different basin types, using multiple types of planes with two dimensions. We demonstrate that both the Jacobi constant as well as the Schwarzschild radius highly influence the character of the orbits, as well as the degree of fractality of the dynamical system.

Symmetric periodic orbits in the Moulton–Copenhagen problem

We consider the planar restricted four-body problem proposed by Moulton. One infinitesimal mass moves under the attraction of three mass points in collinear Euler’s configuration, namely $$P_0 (m_0 = \mu \,m)$$ is placed at the origin, and other two identical points $$P_1 (m)$$ and $$P_2 (m)$$ are placed at the same distance from the origin. The problem is an extension of the well-known Copenhagen problem, in which $$P_0$$ does not exist, and therefore, the name is chosen for the considered problem. We perform a study on the evolution of families of symmetric periodic orbits (characteristic curves) as the mass parameter $$\mu $$ evolves. Compared with the Copenhagen problem, we find new families of periodic orbits and how the classical ones change. We also analyse the number and evolution of spiral points, which represent the heteroclinic orbits connecting equilibrium points.

Motion around the out-of-plane equilibrium points in the photogravitational Copenhagen elliptic restricted three-body problem with oblateness

This paper studies the motion of an infinitesimal mass near the out-of-plane equilibrium points (OEPs) in the elliptic restricted three-body problem (ER3BP) in the case of two equally heavy bodies (Copenhagen problem) where one of the two primaries is a radiation source and the other an oblate spheroid. We found, as in the photogravitational circular restricted three body problem, that the equations of motion of the three dimensional photogravitational ER3BP allow the existence of OEPs if the radiation parameter has negative values. There are two out of plane equilibria that lie in the (ξ, ζ) plane in symmetrical positions with respect to the (ξ, η) plane. The positions of the OEPs are affected by the parameters involved in the systems' dynamics. In particular, the positions change with increase in the radiation pressure, oblateness, eccentricity and semi-major axis of the orbits. As an application, the positions and linear stability of the problem are investigated numerically for the binary system B1534+12. The OEPs are found unstable.

Spatial periodic orbits in the equilateral circular restricted four-body problem: computer-assisted proofs of existence

We use validated numerical methods to prove the existence of spatial periodic orbits in the equilateral restricted four-body problem. We study each of the vertical Lyapunov families (up to symmetry) in the triple Copenhagen problem, as well as some halo and axial families bifurcating from planar Lyapunov families. We consider the system with both equal and non-equal masses. Our method is constructive and non-perturbative, being based on a posteriori analysis of a certain nonlinear operator equation in the neighborhood of a suitable approximate solution. The approximation is via piecewise Chebyshev series with coefficients in a Banach space of rapidly decaying sequences. As by-product of the proof, we obtain useful quantitative information about the location and regularity of the solution.

On the classification of orbits in the three-dimensional Copenhagen problem with oblate primaries

The character of motion for the three-dimensional circular restricted three-body problem with oblate primaries is investigated. The orbits of the test particle are classified into four types: non-escaping regular orbits around the primaries, trapped chaotic (or sticky) orbits, escaping orbits that pass over the Lagrange saddle points L2 and L3, and orbits that lead the test particle to collide with one of the primary bodies. We numerically explore the motion of the test particle by presenting color-coded diagrams, where the initial conditions are mapped to the orbit type and studied as a function of the total orbital energy, the initial value of the z-coordinate and the oblateness coefficient. The fraction of the collision orbits, measured on the color-coded diagrams, show an algebraic dependence on the oblateness coefficient, which can be derived by simple arguments.

Fractal basins of convergence of libration points in the planar Copenhagen problem with a repulsive quasi-homogeneous Manev-type potential

The Newton–Raphson basins of convergence, corresponding to the coplanar libration points (which act as attractors), are unveiled in the Copenhagen problem, where instead of the Newtonian potential and forces, a quasi-homogeneous potential created by two primaries is considered. The multivariate version of the Newton–Raphson iterative scheme is used to reveal the attracting domain associated with the libration points on various type of two-dimensional configuration planes. The correlations between the basins of convergence and the corresponding required number of iterations are also presented and discussed in detail. The present numerical analysis reveals that the evolution of the attracting domains in this dynamical system is very complicated, however, it is a worth studying issue.

On the Newton–Raphson basins of convergence of the out-of-plane equilibrium points in the Copenhagen problem with oblate primaries

The Copenhagen case of the circular restricted three-body problem with oblate primary bodies is numerically investigated by exploring the Newton–Raphson basins of convergence, related to the out-of-plane equilibrium points. The evolution of the position of the libration points is determined, as a function of the value of the oblateness coefficient. The attracting regions, on several types of two-dimensional planes, are revealed by using the multivariate Newton–Raphson iterative method. We perform a systematic and thorough investigation in an attempt to understand how the oblateness coefficient affects the geometry of the basins of convergence. The convergence regions are also related with the required number of iterations and also with the corresponding probability distributions. The degree of the fractality is also determined by calculating the fractal dimension and the basin entropy of the convergence planes.

The Copenhagen problem with a quasi-homogeneous potential

The Copenhagen problem is a well-known case of the famous restricted three-body problem. In this work instead of considering Newtonian potentials and forces we assume that the two primaries create a quasi-homogeneous potential, which means that we insert to the inverse square law of gravitation an inverse cube corrective term in order to approximate various phenomena as the radiation pressure of the primaries or the non-sphericity of them. Based on this new consideration we investigate the equilibrium locations of the small body and their parametric dependence, as well as the zero-velocity curves and surfaces for the planar motion, and the evolution of the regions where this motion is permitted when the Jacobian constant varies.

Determining the Newton-Raphson basins of attraction in the electromagnetic Copenhagen problem

The Copenhagen problem where the primaries of equal masses are magnetic dipoles is used in order to determine the Newton-Raphson basins of attraction associated with the equilibrium points. The parametric variation of the position as well as of the stability of the Lagrange points are monitored when the value of the ratio λ of the magnetic moments varies in predefined intervals. The regions on the configuration (x,y) plane occupied by the basins of convergence are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the basins of attraction of the libration points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We perform a thorough and systematic numerical investigation by demonstrating how the dynamical quantity λ influences the shape, the geometry and also the degree of fractality of the attracting domains. Our numerical results strongly indicate that the ratio λ is indeed a very influential parameter in the electromagnetic binary system.

Crash test for the Copenhagen problem with oblateness

The case of the planar circular restricted three-body problem where one of the two primaries is an oblate spheroid is investigated. We conduct a thorough numerical analysis on the phase space mixing by classifying initial conditions of orbits and distinguishing between three types of motion: (i) bounded, (ii) escape and (iii) collisional. The presented outcomes reveal the high complexity of this dynamical system. Furthermore, our numerical analysis shows a strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. Interpreting the collisional motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We also determined the escape and collisional basins and computed the corresponding escape/crash times. The highly fractal basin boundaries observed are related with high sensitivity to initial conditions thus implying an uncertainty between escape solutions which evolve to different regions of the phase space. We hope our contribution to be useful for a further understanding of the escape and crash mechanism of orbits in this version of the restricted three-body problem.

Basins of Attraction in the Copenhagen Problem Where the Primaries Are Magnetic Dipoles

We deal with the Copenhagen problem where the two big bodies of equal masses are also magnetic dipoles and we study some aspects of the dynamics of a charged particle which moves in the electromagnetic field produced by the primaries. We investigate the equilibrium positions of the particle and their parametric variations, as well as the basins of attraction for various numerical methods and various values of the parameter λ.

On some new aspects of the photo-gravitational Copenhagen problem

We deal with some new aspects of the photo-gravitational Copenhagen case of the restricted three-body problem; more particularly, the distribution and the attracting domains of the stationary solutions of small particles that move in the neighborhood of two major bodies with equal masses when one or both primaries are radiation sources with constant luminosity. Under these conditions, each particle is subjected not only to gravitational forces but to the radiation emitted from the primaries as well.

Asymmetric periodic orbits in the photogravitational Copenhagen problem

In this paper we study the O x -asymmetric solutions of the planar photogravitational restricted three-body problem in the case of primaries with equal masses and equal values of the radiation pressure parameters. In particular, we are concerned with the families of asymmetric orbits which bifurcate from the well known families a, b, and c. Their evolution is examined via the numerical construction of series of the critical bifurcation points of a, b, and c with respect to the variation of the common radiation parameter q . We also present some illustrative cases of these families for several values of this parameter. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, the equations of motion of the problem are regularized by using the Levi-Civita transformations.

Qualitative analysis of the ( N + 1)-body ring problem

In this paper we present a complete study of the ( N + 1)-body ring problem. In particular, we review and describe the evolution of the equilibrium points, their stability, their bifurcations, the zero velocity curves and we provide new techniques that give new views to this classical problem. Some of these techniques are the OFLI2 (a Chaos Indicator given in [Barrio R. Sensitivity tools vs. Poincaré sections. Chaos, Solitons & Fractals 2005;25(3):711–26; Barrio R. Painting chaos: a gallery of sensitivity plots of classical problems. Int J Bifur Chaos Appl Sci Eng [in press]]) and the Crash Test [Nagler J. Crash test for the restricted three-body problem. Phys Rev E (3) 2005;71(2):026227, 11; Nagler J. Crash test for the Copenhagen problem. Phys Rev E (3) 2004;69(6):066218, 6]. With the OFLI2 we have studied the chaoticity of the orbits and with the Crash Test we have classified the orbits as bounded, escape or collisions. Finally, we have performed a systematic search of symmetric periodic orbits of the system, locating much more orbits that in previous studies of other authors.

Crash test for the Copenhagen problem

The Copenhagen problem is a simple model in celestial mechanics. It serves to investigate the behavior of a small body under the gravitational influence of two equally heavy primary bodies. We present a partition of orbits into classes of various kinds of regular motion, chaotic motion, escape and crash. Collisions of the small body onto one of the primaries turn out to be unexpectedly frequent, and their probability displays a scale-free dependence on the size of the primaries. The analysis reveals a high degree of complexity so that long term prediction may become a formidable task. Moreover, we link the results to chaotic scattering theory and the theory of leaking Hamiltonian systems.

Hepatocyte growth factor in human milk and reproductive tract fluids.

Despite evidence indicating a role for hepatocyte growth factor (HGF) in gastrointestinal and reproductive physiology, the concentration and distribution of HGF in human breast milk (BM) and reproductive tract fluids remain unknown. Using enzyme-linked immunosorbent assay (ELISA), the HGF concentrations were determined in human oviductal fluid (hOF), follicular fluid (FF), amniotic fluid (AF), seminal plasma (SP), and colostrum/milk samples, and expression of HGF mRNA by milk cells and AF cells were examined by reverse transcriptase-polymerase chain reaction (RT-PCR). HGF is present at nearly 70-fold normal serum (0.85+/-0.15 ng/mL) concentration in FF (n = 3; x = 57+/-16 ng/mL) and AF (n = 17; x = 57+/-26 ng/mL), and is also present in hOF (n = 3; x = 4.8+/-2.3 ng/mL) and CVL (n = 8; x = 0.7+/-1.1 ng/mL) varying throughout the menstrual cycle. HGF is found at 3-times serum concentration in BM (n = 24; x = 2.3+/-1.3 ng/mL) with no significant difference between premature and full term or stage of lactation (colostrum, transitional, mature milk). HGF mRNA was detected in BM cells but not in AF cells. HGF is present in sufficient amounts to profoundly affect gastrointestinal maturation in the fetus via swallowed AF and neonate via BM, and helps to explain the increased rate of necrotizing enterocolitis (NEC) in infants of premature rupture of membrane (PROM)-complicated pregnancies, and the decreased rate in breast fed neonates. HGF in FF may be necessary for the development and maturation of the oocyte. HGF in hOF, SP, and cervicovaginal lavage (CVL) is likely to enhances epithelial cell integrity and the mucosal barrier. Thus, HGF is widely available in the reproductive tract with functions that remain to be fully elucidated.