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«Astrophysics, Chaos and Complexity in ODED REGEV Department of Physics, Technion-Israel Institute of Technology Department of Astronomy, Columbia ...»

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Astrophysics, Chaos and Complexity in

ODED REGEV

Department of Physics, Technion-Israel Institute of Technology

Department of Astronomy, Columbia University, New York

to appear in: Springer Encyclopedia of Complexity and System Science

http://refworks.springer.com/complexity/

Article Outline

I. Definition of the Subject and its Importance

II. Introduction

III. Hamiltonian Chaos in Planetary, Stellar and Galactic Dynamics

IV. Chaotic Time Variability of Astronomical Sources

V. Spatio-temporal Patterns and Complexity in Extended Systems VI. Future Directions VII. Bibliography I. Definition of the Subject and its Importance Astronomy is the science that deals with the origin, evolution, composition, distance to, and motion of all bodies and scattered matter in the universe. It includes Astrophysics, which is usually considered to be the theoretical part of Astronomy, and as such focuses on the physical properties and structure of cosmic bodies, scattered matter and the universe as a whole. Astrophysics exploits the knowledge acquired in Physics and employs the latter’s methods, in an effort to model astronomical systems and understand the processes taking place in them.

In recent decades a new approach to nonlinear dynamical systems (DS) has been introduced and applied to a variety of disciplines in which DS are used as mathematical models. The theory of chaos and complexity, as this approach is often called, evolved from the study of diverse DS which behave unpredictably and exhibit complex characteristics, despite their seeming simplicity and deterministic nature. The complex behavior is attributed to the property of sensitivity to initial conditions (SIC), whereby despite their completely deterministic behavior, two identical systems in initial states differing only by a minute amount, relatively rapidly develop in very different ways. The theory has been most fruitful in Physics, but it is important to note that its first paradigms were actually deeply related to astrophysical systems.

A large majority of astrophysical systems are theoretically modelled by nonlinear DS. The application of the ideas and methods of chaos and complexity theory to Astrophysics seems thus to be natural. Indeed, these methods have already been exploited in the study of some systems on a vast variety of scales - from planetary satellites through pulsating stars and up to the large scale structure of the universe The main importance of these approaches is in their ability to provide new analytical insights into the intricate physical processes taking place in cosmic matter, shaping the various astronomical objects and causing the prominent observable phenomena that occur in them.

II. Introduction Newton’s laws of mechanics and gravitation had their most prominent application in the significant work of Laplace, Celestial Mechanics. It appeared in five volumes between 1798 and 1827 and summarized his mathematical development and application of Newton’s work. Laplace offered a complete mechanical interpretation of the solar system - planets and their satellites, including the effects of perturbations and tidal interactions. This work had immediately been adopted as an unequivocal manifestation of Nature’s determinism, that is, the possibility to precisely determine the future of any DS - a set of ordinary differential equations (ODE) in this case - if only appropriate initial conditions are known. Laplace himself was so confident in his results that, according to one story, when Napoleon asked him what is the role of the Creator in his theory, he replied that ”this hypothesis” was redundant.

Remarkably, the same system in celestial mechanics (and actually its simplest form

- containing just 3 bodies) which had been considered to be the primary paradigm of determinism gave rise, almost a century after Laplace’s work, to profound analytical difficulties. Another great French mathematician, Henri Poincar´, made e the fundamental discovery that the gravitational n-body system is non-integrable, that is, its general solution can not be expressed analytically, already for n 2.

This work initiated the development of the theory of chaos in Hamiltonian systems, which culminated, about 50 years ago, in the Kolmogorov-Arnold-Moser (KAM) theorem. Section III of this article reviews the application of this theory to planetary, stellar and galactic dynamics.

In unrelated investigations of dissipative DS (Hamiltonian systems are conservative) in the 1960s, aperiodic behavior was observed in numerical studies of simplistic models of thermal convection in geophysical and astrophysical contexts by Lorentz (1963) and Moore & Spiegel (1966). It was conjectured that this behavior is due to SIC, with which these systems were endowed. By the same time, the American mathematician Stephen Smale discovered a new class of ”strange” attractors on which the dynamics is chaotic and which naturally arise in DS sets of ODE or iterated maps (IM) - endowed with SIC. In the early 1980s it was realized, following the work of Mitchell Feigenbaum and others on bifurcations in quadratic IM, that such DS possess a universal behavior as they approach chaos.

Some applications of these developments to astrophysical systems are discussed in Section IV.

A variety of important physical (and astrophysical) systems, notably fluids, are modelled by partial differential equations (PDE). This class of DS provides the spatio-temporal information, necessary for the description and understanding of the relevant phenomena. These DS, especially when they are nonlinear (as is often the case), are significantly more complicated than ODE and IM, but appropriate reduction techniques can sometimes help in applying to them the knowledge gained for ODE and IM. These and a variety of other techniques constitute what is called today pattern theory. Spatio-temporal patterns have been identified and categorized and they help to understand, using analytical and semi-analytical (usually perturbative) approaches, some essential properties of the solutions. It is of interest to properly quantify, describe and understand the processes that create spatio-temporal complexity, like the shapes of interstellar clouds, patterns in thermal convection, turbulent flows and other astrophysically relevant topics. Section V is devoted to a review of these issues.





The quest for the universal and the generic, from which an understanding of complicated and seemingly erratic and heterogenous natural phenomena can emerge, is central to the applications of the theory of chaos, patterns and complexity in DS to Astrophysics. It may provide a powerful investigative tool supplementing fully fledged numerical simulations, which have traditionally been employed in the study of nonlinear DS in Astrophysics III. Hamiltonian Chaos in Planetary, Stellar and Galactic Dynamics Planetary systems, star clusters of various richness, galaxies and galaxy clusters and super-clusters (clusters of clusters) can be approached most directly by considering the gravitational n-body problem (GNBP). In this approach all other interactions, save the gravitational one, are neglected, and the bodies are considered to be point masses. This model is viable, at least approximately, when the distances between the bodies are much large than their typical size and any scattered matter between the bodies has only a very small effect on the dynamics.

The GNBP is one of the paradigms of classical dynamics, whose development originated in Newton’s laws and through the implementation of techniques of mathematical analysis (mainly by Lagrange in his work Analytical Mechanics, published in 1788) finally acquired a powerful

Abstract

formulation in the form of Hamiltonian canonical formalism. Hamiltonian systems, that is, those that obey the Hamilton equations are endowed with important conservation properties linked to symmetries of the Hamiltonian, a function that completely describes the system. This function is defined on the configuration space which consists of the phase space (spanned by the generalized coordinates and momenta) and the time coordinate. A Hamiltonian system conserves phase volume and sometimes also other quantities, notably the total energy (when the Hamiltonian does not depend explicitly on time, as is the case in the GNBP).

The Hamilton equations consist of two first ODE for each degree of freedom, thus the GNBP in a three-dimensional physical space yields, in general, 6n equations and so for large n it is quite formidable. The n = 2 case is reducible to an equivalent one body problem, known as the Kepler problem. The complete solution of this problem was first given by Johann Bernoulli in 1710, quite long before the Lagrange-Hamilton formalism was introduced. The gravitational two-body problem has been successfully applied to various astrophysical systems, e.g. the motion of planets and their satellites and the dynamics of binary stars. The quest for a similar reduction for systems with n 2, was immediately undertaken by several great mathematicians of the time. The side benefit of these studies, conducted for almost two centuries, was a significant progress in mathematics (mainly in the theory of ODE), but definite answers were found in only some particular rather limited cases. Attempts to treat even the simplest problem of this kind, (the restricted, planar, three-body, i.e, where one of the bodies is so light Chaos and Complexity in Astrophysics. Figure 1 Schematic illustration of the homoclinic tangle of the unstable saddle fixed point U. W s and W u are, respectively, the stable and unstable invariant sets (manifolds) of the saddle point. A transversal intersection of these manifolds (ξ0 ) necessarily leads to an infinite number of subsequent intersections, resulting in chaotic behavior of the DS.

that its gravity has no effect on the dynamics and all the orbits are restricted to a plane) ended in a failure. All general perturbative approaches invariably led to diverging terms because of the appearance of ”small divisors” in the perturbation series.

The first real breakthrough came only in the 1880s, when Henri Poincar´ worked e on the GNBP, set for a prize by King Oscar II of Sweden. Poincar´ did not provide e a solution of the problem, but he managed to understand why it is so hard to solve.

By ingenious geometrical arguments, he showed that the orbits in the restricted three-body problem are too complicated to be described by any explicit formula.

In more technical terms, Poincar´ showed that the restricted three body problem e and therefore the general GNBP is non-integrable. He did so by introducing a novel idea, now called a Poincar´ section, with the help of which he was able e to visualize the essentials of the dynamics by means of a two-dimensional area preserving IM. Figure 1 shows schematically a typical mathematical structure, a homoclinic tangle in this case, that is behind chaotic behavior in systems like the restricted three body problem (and, incidentally, also the forced pendulum).

The existence of a hyperbolic fixed point in the appropriate equivalent IM and the transversal intersection of its stable and unstable manifolds gives rise to the complex behavior. Remarkably, Poincar´ was able to visualize such a structure e without the aid of computer graphics.

The non-integrability of the GNBP naturally prompted the question of the Solar system stability. No definite answer to this problem, could however be reasonably expected on the basis of analytical work alone. Before electronic computers became available, this work had largely been based on perturbation or mean field methods.

Despite the difficulties, these efforts yielded new and deep insights on chaotic behavior in Hamiltonian systems after intensive work of close to 50 years. First and foremost among these is the KAM theorem, which elucidated the mathematical process by which an integrable Hamiltonian system transits to chaotic behavior by losing its integrability, when a suitably defined control parameter (e.g. the relative size of the non-integrable perturbation to an integrable Hamiltonian) is gradually increased from zero. In particular, the crucial role of resonant tori in this process has been recognized, the transition starting with the increasing distortion of these tori. The resonant tori become ultimately corrugated on all scales, acquiring a fractal shape and allowing orbits to break-up from them and diffuse in the regions between the surviving non-resonant tori. Over 20 years after the final formulation of the KAM theorem, Boris Chirikov suggested a diagnostic criterion for the onset of chaos in Hamiltonian systems. He studied numerically the standard map Ij+1 = Ij + K sin Θj Θj+1 = Θj + Ij, (1) a particular area preserving two-dimensional IM, where I and Θ are typically action-angle variables of a Hamiltonian system and K is a constant. He showed that fully-spread chaos arises when K 1 and found that this happens because resonances overlap. This, is schematically illustrated in Figure 2.

The basic defining property of deterministic chaos is SIC (due to the divergence of initially arbitrarily close phase space trajectories of the DS) and it is quantied by the positivity of the largest Liapunov exponent, which guarantees chaotic behavior in all DS. In Hamiltonian systems, additional subdivision of different degrees of chaotic motion is available. As the Hamiltonian DS is driven farther into the chaotic regime the motion becomes more strongly irregular, in the sense that the DS trajectories in phase space explore progressively larger volumes. All Hamiltonian DS are recurrent, i.e., their trajectories return infinitely many times arbitrarily close to the initial point (according to the Poincar´ recurrence theorem e this is true for almost all orbits). A DS is called ergodic when long-time averages of a variable is equivalent to its phase-space average. Hamiltonian system are ergodic on non-resonant tori. If any initial phase space volume eventually spreads over the whole space, the then chaotic system is said to have the property of mixing. Deeper into the chaotic regime a Hamiltonian system becomes a K-system. This happens when trajectories in a connected neighborhood diverge exponentially on the average and finally, when every trajectory has a positive Liapunov exponent, there is global instability and the system is called a C-system.



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