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.     

Predictability in dynamical systems

The problem has basically three aspects: A mathematical, a physical and thirdly, an epistemological one. Even though the state of a dynamical system as a function of time may be perfectly and unambiguously determined in principle, do we at this time have the means to find out what this state is?     

Mapping and dynamical systems

This paper reviews various mapping techniques used in dynamical astronomy. It is mostly dealing with symplectic mappings. It is shown that used mappings can be usually interpreted as symplectic integrators. It is not necessary to introduce any functions it is just sufficient to split Hamiltonian into integrable parts. Actually it may be shown that exact mapping with function in the Hamiltonian may be non-symplectic. The application to the study of asteroid belt is emphasised but the possible use of mapping in planetary evolution studies, cometary and other problems is shortly discussed.     

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.     

Strongly perturbed quasi-periodic dynamical systems

It is almost impossible to construct a general theory of the motion of a strongly perturbed dynamical system using classical perturbation theory because this approach uses a reference orbit (e.g. a Keplerian ellipse) which is very different from the actual orbit.A general method, pioneered by Jefferys, is presented here. This method allows each quasi-periodic orbit (for instance a strongly perturbed two body problem: JVIII is the typical example) to specify the coordinates to be used. These coordinates are discovered by a truncated infinite series of coordinate transformations. The transformations are implemented using the idea that the nature of a dynamical system is embodied in the symplectic form. The method is illustracted by a simple example.With modern algebraic and series manipulation languages on present day computers all one needs to begin using this approach is a good numerical integration, the end product being a series for each coordinate. Further weak perturbations are easily incorporated into this semi-analytical solution by the usual methods.Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

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.     

Numerical treatment of non-integrable dynamical systems

A systematic and detailed discussion of the gravitational spring-pendulum problem is given for the first time. A procedure is developed for the numerical treatment of non-integrable dynamical systems which possess certain properties in common with the gravitational problem. The technique is important because, in contrast to previous studies, it discloses completely the structure of two-dimensional periodic motion by examining the stability of the one-dimensional periodic motion. Through the parameters of this stability, points have been predicted from which the one-dimensional motion bifurcates into two-dimensional motion. Consequently, families of two-dimensional periodic solutions emanated from these points are studied. These families constitute the generators of the mesh of all the families of periodic solutions of the problem.     

Irregular extrasolar systems and dynamical instability

The astrometric search for extrasolar planetary systems can regard, at least in the first stage, only systems with high mass ratio (10^–2) of the largest planets to the central star. The formation of such systems is probably a common occurrence in the galaxy, as indicated by various theoretical arguments; these systems could have evolved in a regular way (near-circular and coplanar orbits, Titius-Bode law, etc.) until the most massive accumulating protoplanet exceeded the critical value of the mass ratio corresponding to dynamical instability. This phenomenon has been extensively investigated in recent years, showing that a three- orN-body system becomes unstable (for relative separations similar to the planetary ones) for >10^–2. Therefore, it seems likely that observations of systems within this range of mass ratio will show the irregular end-products of processes related to the instability (close encounters or ejections), which affected drastically the orbital configuration in the last phase of the planet formation.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Effect of dynamical friction on the escape of a supermassive black hole from a Galaxy

We have used the impulsive approximation technique to numerically estimate the effect of dynamical friction on the motion of a supermassive black hole (mass 10⁹ M ) through a galaxy (mass=10¹¹ M ) which has recoiled from the center of the latter as a result of anisotropic emission of gravitational radiation or asymmetric plasma emission. We find the effect to be minimal for recoil taking place at a velocity larger than that of escape at the center of the galaxy. There is a certain critical velocity of ejection (slightly larger than the central escape velocity) at which the black hole must be ejected for the recoil to be successful. Otherwise, dynamical friction becomes relatively pronounced and damped oscillatory motion of the black hole in the potential well of the galaxy ensues. The phenomenon of high-velocity recoil although rare, can be astrophysically spectacular in view of the fact that the black hole would carry a substantial amount of gaseous material as well as a very large number of galactic stars. Some recent observations are cited where the recoil phenomenon might be applicable.     

Family boundary curves for autonomous dynamical systems

The notion of the family boundary curves (FBC), introduced recently for two-dimensional conservative systems, is extended to account for, generally, nonconservative autonomous systems of two degrees of freedom. Formulae are found for the force componentsX (x, y),Y (x, y) which produce a preassigned family of orbitsf(x, y)=c lying inside a preassigned, open or closed, regionB(x, y)0 of the xy plane.     

The dynamical decay of unstable 4-body systems

Unstable 4-body systems with negative energy can ultimately decay to (1) a binary plus two single stars, (2) two separate binaries, or (3) a stable triple plus a single star. One hundred random 2-dimensional and one hundred random 3-dimensional 4-body systems have been numerically integrated to determine the statistics of the end products. Of the final stable triples and binaries, 19% were triples, which agrees well with observational estimates of the ratio of triples to binaries. The results were essentially the same for 2- and 3- dimensional systems.     

Orbital structure of self-consistent triaxial stellar systems

We used a multipolar code to create, through the dissipationless collapses of systems of 1,000,000 particles, three self-consistent triaxial stellar systems with axial ratios corresponding to those of E4, E5 and E6 galaxies. The E5 and E6 models have small, but significant, rotational velocities although their total angular momenta are zero, that is, they exhibit figure rotation; the rotational velocity decreases with decreasing flattening of the models and for the E4 model it is essentially zero. Except for minor changes, probably caused by unavoidable relaxation effects, the systems are highly stable. The potential of each system was subsequently approximated with interpolating formulae yielding smooth potentials, stationary for the non-rotating model and stationary in the rotating frame for the rotating ones. The Lyapunov exponents could then be computed for randomly selected samples of the bodies that make up the different systems, allowing the recognition of regular and partially and fully chaotic orbits. Finally, the regular orbits were Fourier analyzed and classified using their locations on the frequency map. As it could be expected, the percentages of chaotic orbits increase with the flattening of the system. As one goes from E6 through E4, the fraction of partially chaotic orbits relative to that of fully chaotic ones increases, with the former surpassing the latter in model E4; the likely cause of this behavior is that triaxiality diminishes from E6 through E4, the latter system being almost axially symmetric. We especulate that some of the partially chaotic orbits may obey a global integral akin to the long axis component of angular momentum. Our results show that is perfectly possible to have highly stable triaxial models with large fractions of chaotic orbits, but such systems cannot have constant axial ratios from center to border: a slightly flattened reservoir of highly chaotic orbits seems to be mandatory for those systems.     

Spectra of stretching numbers and helicity angles in dynamical systems

We define a stretching number (or Lyapunov characteristic number for one period) (or stretching number) a = In % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada% Wcaaqaaiabe67a4jaadshacqGHRaWkcaaIXaaabaGaeqOVdGNaamiD% aaaaaiaawEa7caGLiWoaaaa!3F1E!\[\left| {\frac{{\xi t + 1}}{{\xi t}}} \right|\]as the logarithm of the ratio of deviations from a given orbit at times t and t + 1. Similarly we define a helicity angle as the angle between the deviation % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaam% iDaaaa!3793!\[\xi t\]and a fixed direction. The distributions of the stretching numbers and helicity angles (spectra) are invariant with respect to initial conditions in a connected chaotic domain. We study such spectra in conservative and dissipative mappings of 2 degrees of freedom and in conservative mappings of 3-degrees of freedom. In 2-D conservative systems we found that the lines of constant stretching number have a fractal form.     

The dynamical stability of W Ursae Majoris-type systems

Theoretical study indicates that a contact binary system would merge into a rapidly rotating single star due to tidal instability when the spin angular momentum of the system is more than a third of its orbital angular momentum. Assuming that W Ursae Majoris (W UMa) contact binary systems rigorously comply with the Roche geometry and the dynamical stability limit is at a contact degree of about 70 per cent, we obtain that W UMa systems might suffer Darwin's instability when their mass ratios are in a region of about 0.076–0.078 and merge into the fast-rotating stars. This suggests that the W UMa systems with mass ratio   q ≤ 0.076  cannot be observed. Meanwhile, we find that the observed W UMa systems with a mass ratio of about 0.077, corresponding to a contact degree of about 86 per cent would suffer tidal instability and merge into the single fast-rotating stars. This suggests that the dynamical stability limit for the observed W UMa systems is higher than the theoretical value, implying that the observed systems have probably suffered the loss of angular momentum due to gravitational wave radiation (GR) or magnetic stellar wind (MSW).     

Numerical integration of dynamical systems with Lie series

The integration of the equations of motion in gravitational dynamical systems—either in our Solar System or for extra-solar planetary systems—being non integrable in the global case, is usually performed by means of numerical integration. Among the different numerical techniques available for solving ordinary differential equations, the numerical integration using Lie series has shown some advantages. In its original form (Hanslmeier and Dvorak, Astron Astrophys 132, 203 1984), it was limited to the N-body problem where only gravitational interactions are taken into account. We present in this paper a generalisation of the method by deriving an expression of the Lie terms when other major forces are considered. As a matter of fact, previous studies have been done but only for objects moving under gravitational attraction. If other perturbations are added, the Lie integrator has to be re-built. In the present work we consider two cases involving position and position-velocity dependent perturbations: relativistic acceleration in the framework of General Relativity and a simplified force for the Yarkovsky effect. A general iteration procedure is applied to derive the Lie series to any order and precision. We then give an application to the integration of the equation of motions for typical Near-Earth objects and planet Mercury.     

Linearization of dynamical systems using integrals of the motion

A method is presented which transforms certain non-linear differential equations of dynamics into linear equations by introducing a new independent variable and by utilizing the integrals of motion. As examples of special interest the linearizations of unperturbed and perturbed Keplerian motions are discussed.