Orbital Distribution Arbitrarily Close to the Homothetic Equilateral Triple Collision in the Free‐Fall Three‐Body Problem with Equal Masses

The existence of escape and nonescape orbits arbitrarily close to the homothetic equilateral triplecollision orbit is considered analytically in the threebody problem with zero initial velocities and equal masses. It is proved that escape orbits in the initial condition space are distributed around three kinds of isosceles orbits. It is also proved that nonescape orbits are distributed in between the escape orbits where different particles escape. In order to show this, it is proved that the homotheticequilateral orbit is isolated from other triplecollision orbits as far as the collision at the first triple encounter is concerned. Moreover, the escape criterion is formulated in the planarisosceles problem and translated into the words of regularizing variables. The result obtained by us explains the orbital structure numerically.     

Motions close to escapes in the rhomboidal four body problem

In this work we study escape and capture orbits in the planar rhomboidal 4-body problem in a level of constant negative energy. There are only two different values of the masses here. By using numerical analysis, we show certain transversal intersections of the invariant manifolds of parabolic orbits. We then introduce Symbolic Dynamics when the mass ratio is small, and when it is close to one. In the first case the escapes or captures predominate in the direction of one of the diagonals of the rhombus, while in the second case we find solutions escaping or being captured in the direction of both possible diagonals.     

Exact Periodic Orbits and Chaos in Polynomial Potentials

Using the theory of the inverse problem, we construct galactic type potentials based on perturbed harmonic oscillators, producing exact analytic trajectories represented by second-order curves. The various families of orbits are examined in these potentials. It is found, that there are cases where our galactic models produce exact analytic orbits and chaotic orbits as well. The coexistence of analytic and chaotic orbits, happens when the energy of the test particle is close to the energy of escape. Comparison with previous work is also made. This revised version was published online in July 2006 with corrections to the Cover Date.     

Oscillatory orbits in the planar three‐body problem with equal masses

In the free‐fall three‐body problem, distributions of escape, binary, and triple collision orbits are obtained. Interpretation of the results leads us to the existence of oscillatory orbits in the planar three‐body problem with equal masses. A scenario to prove their existence is described. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of Two-Parameter Families of Triple Close ApproachesOccurring in Stellar Systems (I)

The analysis of two-parameter families of triple close approaches occurring in stellar systems is studied in a series of three papers. This paper deals with the role of triple encounters with low initial velocities and equal masses in the evolution of stellar systems. It shows how a condition of complete collapse may be perturbed to obtain well-established two-parameter families of asymmetric triple close approaches with the formation of a binary and with systematic regularity of escape of the third body. Our results also indicate that the conjecture of Szebehely viz., `The measure of escaping orbits is significantly higher than the measure of stable orbits' is likely to be true. Further our results differ from that of Agekian's escape probability criterian. The second paper deals with the role of triple encounters with low initial velocities and equal masses in the evolution of stellar systems in 3D space. The third and last paper offers applications in stellar systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Escapes and Recurrence in a Simple Hamiltonian System

Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincaré recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite minimum recurrence time. We find the minimum recurrence time as a function of the energy.This revised version was published online in October 2005 with corrections to the Cover Date.     

Analysis of impulsive maneuvers to keep orbits around the asteroid 2001SN<Subscript>263</Subscript>

The strongly perturbed environment of a small body, such as an asteroid, can complicate the prediction of orbits used for close proximity operations. Inaccurate predictions may make the spacecraft collide with the asteroid or escape to the deep space. The main forces acting in the dynamics come from the solar radiation pressure and from the body’s weak gravity field. This paper investigates the feasibility of using bi-impulsive maneuvers to avoid the aforementioned non-desired phenomena (collisions and escapes) by connecting orbits around the triple system asteroid 2001SN₂₆₃, which is the target of a proposed Brazilian space mission. In terms of a mathematical formulation, a recently presented rotating dipole model is considered with oblateness in both primaries. In addition, a “two-point boundary value problem” is solved to find a proper transfer trajectory. The results presented here give support to identifying the best strategy to find orbits for close proximity operations, in terms of long orbital lifetimes and low delta-\(V\) consumptions. Numerical results have also demonstrated the significant influence of the spacecraft orbital elements (semi-major axis and eccentricity), angular position of the Sun and spacecraft area-to-mass ratio, in the performance of the bi-impulsive maneuver.     

Triple collisions in the one-dimensional three-body problem

We consider the motions of particles in the one-dimensional Newtonian three-body problem as a function of initial values. Using a mapping of orbits to symbol sequences we locate the initial values leading to triple collisions. These turn out to form curves which give clear structure to the region in which the motions depend sensitively on initial conditions. In addition to finding the triple collision orbits we also locate orbits which end up to a triple collision in both directions of time, that is, orbits which are finite both in space and time. This revised version was published online in July 2006 with corrections to the Cover Date.     

Regular and chaotic dynamics in 3D reconnecting current sheets

We consider the possibility of particles being injected at the interior of a reconnecting current sheet (RCS), and study their orbits by dynamical systems methods. As an example we consider orbits in a 3D Harris type RCS. We find that, despite the presence of a strong electric field, a 'mirror' trapping effect persists, to a certain extent, for orbits with appropriate initial conditions within the sheet. The mirror effect is stronger for electrons than for protons. In summary, three types of orbits are distinguished: (i) chaotic orbits leading to escape by stochastic acceleration, (ii) regular orbits leading to escape along the field lines of the reconnecting magnetic component, and (iii) mirror-type regular orbits that are trapped in the sheet, making mirror oscillations. Dynamically, the latter orbits lie on a set of invariant KAM tori that occupy a considerable amount of the phase space of the motion of the particles. We also observe the phenomenon of 'stickiness', namely chaotic orbits that remain trapped in the sheet for a considerable time. A trapping domain, related to the boundary of mirror motions in velocity space, is calculated analytically. Analytical formulae are derived for the kinetic energy gain in regular or chaotic escaping orbits. The analytical results are compared with numerical simulations.     

Analysis of Families of Triple Close Approaches Occurring inStellar Systems in Three Dimensions (II)

This is the second paper of a trilogy dealing with the role of triple encounters with low initial velocities and equal masses in the evolution of stellar systems in three dimensional space. It shows how a condition of complete collapse may be perturbed to obtain well-established families of asymmetric triple close approaches with systematic regularity of escape with the formation of a binary. The main result is that when perturbation is introduced two close approaches called the first close approach and the second close approach occur in the same plane but the binary formed and the escaper are not in that plane. Further it is observed that the conjecture of Szebehely (1977) viz. `The measure of escaping orbits is significantly higher than the measure of stable orbits' is likely to be true. The third and last paper offers applications in stellar systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Search for Collision Orbits in the Free-Fall Three-Body Problem II

A numerical procedure to systematically find collision orbits in the planar three-body problem has been developed in the preceding paper (Tanikawa et al., 1995). Using this procedure, a search for binary and triple collision orbits has been carried out in the free-fall three-body problem. Some detailed structures of a part of the initial value space are discussed. Various interesting orbits have been found. Examples are oscillatory orbits in which ejected particles change from ejection to ejection, and orbits which are not isosceles initially but nearly isosceles after escape. Some results of isosceles problems (Simó and Martínez, 1988) are extended to non-isosceles problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Main features of dynamical escape from three-dimensional triple systems

The dynamical evolution of triple systems with equal and unequal-mass components and different initial velocities is studied. It is shown that, in general, the statistical results for the planar and three-dimensional triple systems do not differ significantly. Most (about 85%) of the systems disrupt; the escape of one component occurs after a triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (about 15%) of stable or long-lived systems is formed if the angular momentum is non-zero. Averages, distributions and coefficients of correlations of evolutionary characteristics are presented: the life-time, angular momentum, numbers of wide and close triple approaches of bodies, relative energy of escapers, minimum perimeter during the last triple approach resulting in escape, elements of orbits of the final binary and escaper.     

Stickiness effects in chaos

Stickiness is a temporary confinement of orbits in a particular region of the phase space before they diffuse to a larger region. In a system of 2-degrees of freedom there are two main types of stickiness (a) stickiness around an island of stability, which is surrounded by cantori with small holes, and (b) stickiness close to the unstable asymptotic curves of unstable periodic orbits, that extend to large distances in the chaotic sea. We consider various factors that affect the time scale of stickiness due to cantori. The overall stickiness (stickiness of the second type) is maximum near the unstable asymptotic curves. An important application of stickiness is in the outer spiral arms of strong-barred spiral galaxies. These spiral arms consist mainly of sticky chaotic orbits. Such orbits may escape to large distances, or to infinity, but because of stickiness they support the spiral arms for very long times.     

Multiple Periodic Orbits in the Hill Problem with Oblate Secondary

We study the multiple periodic orbits of Hill’s problem with oblate secondary. In particular, the network of families of double and triple symmetric periodic orbits is determined numerically for an arbitrary value of the oblateness coefficient of the secondary. The stability of the families is computed and critical orbits are determined. Attention is paid to the critical orbits at which families of non-symmetric periodic orbits bifurcate from the families of symmetric periodic orbits. Six such bifurcations are found, one for double-periodic and five for triple-periodic orbits. Critical orbits at which families of sub-multiple symmetric periodic orbits bifurcate are also discussed. Finally, we present the full network of families of multiple periodic orbits (up to multiplicity 12) together with the parts of the space of initial conditions corresponding to escape and collision orbits, obtaining a global view of the orbital behavior of this model problem.     

Transition spectra of dynamical systems

The spectra of ‘stretching numbers’ (or ‘local Lyapunov characteristic numbers’) are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima correspond to maxima or minima in the curve a(θ), where a is the stretching number, and θ the azimuthal angle. We explain the appearance of new maxima in the spectra of ordered orbits. The orbits just outside the last KAM curve are confined close to this curve for a long time (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered orbits just inside the last KAM curve, but later these spectra tend to the invariant spectrum of the chaotic domain. The sticky spectra are invariant during the stickiness time. The stickiness time increases exponentially as we approach an island of stability, but very close to an island the increase is super exponential. The stickiness time varies substantially for nearby orbits; thus we define a probability of escape P_n(x) at time n for every point x. Only the average escape time in a not very small interval Δx around each x is reliable. Then we study the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spectrum within a given accuracy. In the regular domain N is small, while in the chaotic domain it is large. In some ordered cases the convergence is anomalously slow. In these cases the maximum value of a_k in the continued fraction expansion of the rotation number a = [a₀,a₁,... a_k,...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calculating orbits starting at various points along a line parallel to the q-axis. A monotonic variation of the sup {q}as a function of the initial condition q₀ indicates ordered motions, a jump indicates the crossing of a localized chaotic domain, and a V-shaped structure indicates the crossing of an island. But sometimes the V-shaped structure disappears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Two Fixed Centers: An Exceptional Integrable System

It is usually believed that we know everything to be known for any separable Hamiltonian system, i.e. an integrable system in which we can separate the variables in some coordinate system (e.g. see Lichtenberg and Lieberman 1992, Regular and Chaotic Dynamics, Springer). However this is not always true, since through the separation the solutions may be found only up to quadratures, a form that might not be particularly useful. A good example is the two-fixed-centers problem. Although its integrability was discovered by Euler in the 18th century, the problem was far from being considered as completely understood. This apparent contradiction stems from the fact that the solutions of the equations of motion in the confocal ellipsoidal coordinates, in which the variables separate, are written in terms of elliptic integrals, so that their properties are not obvious at first sight. In this paper we classify the trajectories according to an exhaustive scheme, comprising both periodic and quasi-periodic ones. We identify the collision orbits (both direct and asymptotic) and find that collision orbits are of complete measure in a 3-D submanifold of the phase space while asymptotically collision orbits are of complete measure in the 4-D phase space. We use a transformation, which regularizes the close approaches and, therefore, enables the numerical integration of collision trajectories (both direct and asymptotic). Finally we give the ratio of oscillation period along the two axes (the ‘rotation number’) as a function of the two integrals of motion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Periodic orbits and escapes in dynamical systems

We study the periodic orbits and the escapes in two different dynamical systems, namely (1) a classical system of two coupled oscillators, and (2) the Manko-Novikov metric which is a perturbation of the Kerr metric (a general relativistic system). We find their simple periodic orbits, their characteristics and their stability. Then we find their ordered and chaotic domains. As the energy goes beyond the escape energy, most chaotic orbits escape. In the first case we consider escapes to infinity, while in the second case we emphasize escapes to the central ??bumpy?? black hole. When the energy reaches its escape value, a particular family of periodic orbits reaches an infinite period and then the family disappears (the orbit escapes). As this family approaches termination it undergoes an infinity of equal period and double period bifurcations at transitions from stability to instability and vice versa. The bifurcating families continue to exist beyond the escape energy. We study the forms of the phase space for various energies, and the statistics of the chaotic and escaping orbits. The proportion of these orbits increases abruptly as the energy goes beyond the escape energy.     

Escape regions of a quartic potential

We investigate the escape regions of a quartic potential and the main types of irregular periodic orbits. Because of the symmetry of the model the zero velocity curve consists of four summetric arcs forming four open channels around the lines y = ± x through which an orbit can escape. Four unstable Lyapunov periodic orbits bridge these openings.We have found an infinite sequence of families of periodic orbits which is the outer boundary of one of the escape regions and several infinite sequences of periodic orbits inside this region that tend to homoclinic and heteroclinic orbits. Some of these sequences of periodic orbits tend to homoclinic orbits starting perpendicularly and ending asymptotically at the x-axis. The other sequences tend to heteroclinic orbits which intersect the x-axis perpendicularly for x > 0 and make infinite oscillations almost parallel to each of the two Lyapunov orbits which correspond to x > 0 or x < 0.     

The Photogravitational Hill Problem: Numerical Exploration

We consider the recently introduced version of the classical Lunar Hill problem, the photogravitational Hill problem, and study it's equilibrium points and zero-velocity curves. The full network of families of periodic orbits is numerically explored, their stability is computed and critical orbits are determined. Non-periodic orbits are also computed as points on a surface of section, providing an outlook of the stability regions, chaotic motions and escape.